O'Keefe, John A.

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https://ntrs.nasa.gov/search.jsp?R=20160000487 2019-08-31T04:50:13+00:00Z
JOURNAL OF GEOPHICAT CH 
N	 100036  
VOLUME 66, No. 6 JUNE 1961 
0 V*
Discussion of Paper by W. A. Heiskanen,
'The Latest Achievements of Physical Geodesy' 
JOHN A. O'KEEFE 
Theoretical Division, Goddard Space Flight Center 
National Aeronautics and Space Administration
Washington, D. C. 
In his recent report on physical geodesy, 
Ileiskanen [1960] replied to a criticism by 
O'Keefe [1959]. He wrote, 'O'Keefe has as-
sumed that the flattening value of 1:299.8 cor-
responds to the hydrostatic equilibrium, and he 
uses this flattening value when he computes the 
spherical harmonics of the gravity anomalies 
and the undulations N of the geoid. Because the 
hydrostatic equilibrium is not complete, one 
cannot use this value, and Figure 2, page 2931, 
of O'Keefe's [1959] publication will not ex-
plain anything.' 
In view of objections raised by Professor 
Heiskanen, it appears necessary to state more 
fully the reasoning leading to the adoption of 
the value 1/299.8 for the flattening of the refer-
ence ellipsoid. The underlying idea is to use as 
reference the figure that the earth would assume 
if it were in a state of fluid equilibrium. The 
advantage of this figure is that the stresses in 
the interior of the earth can be expected to be 
proportional to the amplitude of harmonics in 
the gravitational field that measure deviations 
from this ellipsoid. For this reason, namely, the 
interest to the geophysicist concerned, with the 
structure of the earth, we have stressed the im-
portance of the figure of fluid equilibrium. 
There is, in addition, significance for geodesy 
in the disclosure that the geoid differs systemat-
ically and extensively from the figure of fluid 
equilibrium. These differences contradict the 
principle formulated by Heiskanen and Vening 
Meinesz [1958] as the basic hypothesis of geod-
esy, namely, that deviations from the ellipsoid 
of fluid equilibrium would be found to be small. 
The actual deviations have been found to be 
between 5 and 30 times those anticipated. For 
this reason a firm adherence to the ellipsoid of 
fluid equilibrium as the reference for calcula-
tions of anomalies is useful to the geodesist by 
keeping before his eyes the real size of the
deviations which may exist in those parts of the 
world that he cannot measure. 
The value 1/299.8 is based on a slight modifi-
cation of the work of Henriksen [1960]. Hen-
riksen's argument is summarized by the block 
diagram, Figure 1 [O'Keefe, Roman, Yaplee, 
and Eckels, 1959], which presents the relations 
among the physical constants related to the 
flattening of the earth. From the precession of 
the equinoxes, the value H = 3.274 x 10' is 
derived for the quantity (C - A)/C. On the 
other hand, from observations of satellites the 
value J2 = 1.0825 x 10 is found for the quan-
tity (C - A )/M&, whence C/Md' = 3.306 [cf. 
Cook, 1959]. The point of Henriksen's paper is 
that the hydrostatic value of the flattening ap-
propriate to a given configuration of the earth 
can be directly calculated from this quantity 
C/Ma'. That this is so is directly evident from 
the first-order equations of Jeffreys [1952], espe-
cially equation 47, from which we can determine 
the quantity 77 , and equation 50, which relates 
to the flattening. These equations constitute 
the heart of the problem. However, a slightly 
more refined treatment can be made either by 
the equation of de Sitter [1924] which Henrik-
sen used or by the equations given by Jones 
[1954]. The numerical result is 1/299.8. 
By way of comparison, the method used be-
fore the advent of the satellites for this purpose 
Fig. 1. Relationship or quantities involved in the 
flattening of the earth. 
1992
LETTERS TO THE EDITOR	 1993 
was somewhat roundabout. Lacking an accurate 
determination of (C - A)/M&, the assumption 
was made that this quantity had precisely the 
value that it would have in the case of fluid 
equilibrium. This equation, together with the 
condition arising from the moon's precession, 
was sufficient to determine the value of the flat-
tening; and in this way the old hydrostatic 
value of 1/297.3 was found [Jeffreys, 1952]. 
Since we now know that the earth is not in 
hydrostatic equilibrium, it is clearly no longer 
legitimate to calculate in this way. 
It may be asked whether we can assume that 
the value of C which we obtained from the ob-
servations is identical with the value which the 
earth would have if it should go to hydrostatic 
equilibrium. The answer is that this is in all 
probability true to four significant figures. A 
change of C or A by one part in 100,000 is 
enough to change (C - A)/M& in the third 
significant figure. Thus the earth can change its 
flattening to agree with the requirements of hy-
drostatic equilibrium without seriously altering 
its moment of inertia. It is these considerations 
that justify the flattening of 1/299.8. It will be 
seen that they take into account fully the fact 
that hydrostatic equilibrium is not complete. 
Heiskanen further asserts that he has 'not 
claimed that complete hydrostatic equilibrium 
prevails.' On this point there is considerable 
agreement. It seems to me, however, that the
new numerical data mean that in estimating the 
contributions of the unexplored portions of the 
earth's gravitational field one should assume 
nonhydrostatic contributions in accordance with 
those found from satellites, i.e., approximately 
10 times the contributions assumed by Heiska-
nen and Vening Meinesz [1958] in their formu-
lation of the basic hypothesis of geodesy. 
REFERENCES 
Cook, A. H., Geophys. J., 2, 222-239, 1959. 
de Sitter, W., Bull. Astr. Inst. Neth., 55, 97-108, 
1924. 
Heiskanen, W. A., The latest achievements of 
physical geodesy, J. Geophys. Research, 65, 
2827-2836, 1960. 
Heiskanen, W. A., and F. A. Vening Meinesz, 
The Earth and Its Gravity Field, McGraw-Hill 
Book Co., pp. 72, 73, 1958.	 - 
Henriksen, S. W., Annals of IGY, 12, 197-198, 
1960. 
Jeffreys, Harold, The Earth, Cambridge University 
Press, 1952. 
Jones, Harold Spencer, The Earth as a Planet, 
edited by G. P. Kuiper, University of Chicago 
Press, pp. 1-41, 1954. 
O'Keefe, J. A., Zonal harmonics of the earth's 
gravitational field and the basic hypothesis of 
geodesy, J. Geophys. Research, 64, 2389-2392, 
1959. 
O'Keefe, J A., Nancy G. Roman, Benjamin S. 
Yaplee, and Ann Eckels, Monograph No. 4, 
American Geophysical Union, p. 50, 1959. 
(Received February 6, 1961.)


The Latest Achievements of Physical Geodesy

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