Geophys. J . Int. (1991) 106, 699-702
RESEARCH NOTE
The Earth's equatorial principal axes and moments of inertia
H. S. Liu and B. F. Chao
Geodynamics Branch, Goddard Space Flight Center, Greenbelt, M D 20771, USA
Accepted 1991 February 26. Received 1990 October 22
s2,
Key words: Earth's moment of inertia, Earth's rotation.
INTRODUCTION
Let the three principal moments of inertia of the Earth be
A, B and C , where A < B < C . Let the corresponding
principal axes be called the a, b and c axes, and the
geographical coordinate axes be x, y and z. Disregarding
internal relative motions, the principal axes are fixed in the
geographical coordinates. Apart from a slight polar motion
in the Earth's rotation, the z axis coincides with the c axis
(Fig. l ) , whose orientation and the corresponding value of C
are well determined. The present paper focuses on the other
two principal axes, which lie on the xy, or equatorial plane,
and their corresponding 'equatorial' principal moments of
inertia. Specifically, where are the a and b axes in the
equatorial plane? And what can one say about A and B?
The study of the Earth as a triaxial body is a classical
problem; the pre-space era history of investigations using
astrogeodetic and gravimetric methods were reviewed by
Izsak (1961) and Heiskanen (1962); the results were crude
and in discord. Progressively better results were obtained
when it was realized that the orientation of the a and b axes
and the difference B - A can be determined from the
Earth's external gravitational field coefficients (the Stokes
coefficients) of degree 2 and order 2, which were inferred
from artificial satellite orbit observations (e.g., Izsak 1961;
Blitzer ef al. 1962). These and subsequent studies that did
discuss, or include (usually in passing) a discussion of the
matter fell short of providing explicit derivation and
expressions (e.g., Goldreich & Toomre 1969; Lorell 1972;
Wagner 1973; Kinoshita 1977; Bills & Ferrari 1978; Kaula
1979; Bills, Kieffer & Jones 1987; Chao & Gross 1987; Chao
& Rubincam 1989). Moreover, as far as we are aware,
textbook coverage of the subject has been skimpy (e.g.,
Heiskanen & Moritz 1967; Baker & Makemson 1967;
Jeffreys 1976; Torge 1980; Hubbard 1984), if treated at all.
Although far less significant than the orientation of the c
axis and the difference C - A , the quantities under present
consideration are not without dynamical implications. Chao
(1989) has pointed out that the orientation of the a and b
axes serves as an important constraint on laterally
heterogeneous density models for the Earth, bearing on
tectonic movements, mantle convections, and perhaps core
configuration and activities. Indeed the difference B - A
(and other Stokes coefficients for that matter) can serve as
such constraints as well. Moreover, Liu & Chao (1990)
argue that the luni-solar torque exerted on the Earth's
rotation owing to the non-zero B - A (e.g., Woolard 1953)
approaches the threshold of detection envisioned for the
near future (cf. Dickey & Schutz 1989). On other
(terrestrial) planets for which data are available, these
effects can be much larger in magnitude although any
geophysical discussion becomes more speculative. At any
rate we believe it is worthwhile to devote a short paper to
the subject.
FORMULA
The external gravitational field of a planetary body,
satisfying the Laplace's equation, is customarily given by
x ( C , cos mA + S, sin mA),
(1)
699
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SUMMARY
Let the Earth's equatorial principal moments of inertia be A and B, where A < B,
and the corresponding principal axes be a and b. Explicit formulae are here derived
for determining the orientation of a and b axes and the difference B - A using CZ2
and
the two gravitational harmonic coefficients of degree 2 and order 2. For
the Earth, the a axis lies along the (14.93"W' 165.07"E) diameter, and the b axis lies
perpendicular to it along the (75.07"E' 104.93"W) diameter. The difference B - A is
MR2. These quantities for other planets are contrasted, and geophysi7.260 x
cal implications are discussed.
700
H . S. Liu and B. F. Chao
z=c
of Z, on gets
C,,
+ is,, = J(x2 - y2 + 2ixy)p d V / K
=l ( x
a
(3)
+
= ( B - A)ei2"/K .
(4)
Equating the real and imaginary parts in equation (4) one
gets an expression for the (positive) difference in the two
equatorial moments of inertia:
B - A = KY-,
(54
and the east longitude of the principal axis of the least
moment of inertia:
A = f arg(C,,
+ is.,).
(5b)
Note that, because of the factor 1/2 in (Sb), it is
arg(C,,+iS,,)
(between -n and n), rather than
tan-'(S,,/C,,)
(between - n / 2 and n/2), that should be
used in equation (5b). For arg(C,,+iS,,)
between - n / 2
and n/2, the two are the same and either gives the correct
A. Otherwise, tan-'(&2/C22) would be off by n/2, hence
switching the a and b axes.
RESULTS A N D DISCUSSION
A complete detailed derivation of the above with explicit
normalization factors can be found in, e.g., Chao & Gross
(1987).
We note in passing that there are five degree-2 Stokes
coefficients but six independent components in Z.
Consequently, in order to determine Z uniquely one needs
an additional piece of information; for instance the
astronomically measured precession constant H = (C A ) / C in the case of the Earth, or the physical longitudinal
libration parameter y = ( B - A ) / C in the case of the Moon.
This, of course, is a manifestation of the classical
non-uniqueness in the gravitational inverse problem (e.g.,
Morse & Feshbach 1953, pp. 1276-1283). In particular,
given C,, and S, one can only obtain the orientation of a
and b in the equatorial plane and the difference B - A , as
follows.
Here the complex notation is found to be extremely
convenient. We will take the xy plane to be the complex
plane, the x and y axes being the real and imaginary axes,
respectively. Then from equation (2d,e) and the definition
The observed Earth values for degree-2 Stokes coefficicnts
of the GEM-TZ model are: C,, = -484.1653(3) X
CZ2= 2.4390(2) X
and
&, = -1.4001(2) X lop6
(Marsh et al. 1990). With the known values of M R 2 and C,
equation (5a) gives
B-A
--
- 7.260 X
MR,
and the longitudinal libration parameter
y = ( B - A ) / C = 2.1946 x
(7)
With known H (see above) one gets ( B - A ) / ( C - A ) =
1/150; that is, the Earth's equatorial dynamical ellipticity is
150 times smaller than the polar dynamical ellipticity.
Equation (7) is about twice as large as the value quoted by
Scheidegger [1982, p. 197, which, incidentally, erroneously
takes (7) to be the cause of the Earth's free wobble;
Scheidegger 1983, personal communication].
Similarly, from equation (5b) and to the accuracy of the
C,, and S, coefficients,
A = -14.93".
(8)
That is, the a axis of the Earth lies along the (14.93"W,
165.07"E) diameter, and the b axis lies perpendicular to it
along the (75.07"E, 104.93"W) diameter. The latter points
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C,, + i&,
where G is the gravitational constant; M and R are the
Earth's mass and mean radius, respectively; r = (r, 0, A)
gives the radial distance, the colatitude and the east
longitude of the field point; and 6, is the 4n-normalized
associated Legendre function of degree I and order m. The
are the (normalized) Stokes
parameters Cf, and S,
coefficients that are now routinely determined for the Earth
from observations of satellite orbits. Comparing equation
(1) with the multiple expansion of U (e.g., Jackson 1975),
one can show that the Stokes coefficients are simply
normalized multipoies of the Earth's mass density
distribution p(r). In particular, the inertia tensor Z of the
Earth is related to the degree-2 Stokes coefficients in the
following manner:
dV/K
where K = 2(5/3)'"MR2 for brevity. Invoking the coordinate rotation in the xy plane through an angle A (Fig. l), the
integrand in equation (3) becomes, in the new coordinates a
and b, ( x iy)' = (a ib)2ei2". Now if the new axes happen
to be the principal axes (hence the use of symbols a and b
here), then by definition the product of inertia $abp dV
vanishes, and the integral in equation (3) becomes
ei2" $(a + ib),p dV = ei2"$(a2 - b2)p dV. Then, by the
definition of A and B,
+
Figure 1. The relation between the principal axes (u, b, c) and the
geographical axes (x, y, 2). Angle A is the east longitude of a, the
principal axis of the least moment of inertia.
+ iy)'p
Earth’s principal axes and moments of inertia
+
accumulate (e.g., Dickey & Schutz 1989). It should be noted
that these amplitudes are not much smaller than the
semi-diurnal ocean tide contributions in the rotational speed
variation (Yoder, Williams & Parke 1981; but see Baader,
Brosche & Hovel 1983). The separation of the two effects in
the observation is therefore desirable in studying the
behaviour of the ocean tides.
ACKNOWLEDGMENTS
We would like to thank B. G. Bills for discussions and
comments.
REFERENCES
Baader, H. R., Brosche, P. & Hovel, 1983. Ocean tides and
periodic variations of the Earths rotation, J . Geophys., 52,
140-142.
Baker, R. M. L. & Makemson, M. W., 1967. Asrrodynamics, 2nd
edn, Academic Press, New York.
Balmino, G., Moynot, B. & Vales, N., 1982. Gravity field model of
Mars in spherical harmonics up to degree and order eighteen,
J. geophys. Res., 87, 9735-9746.
Bills, B. G. & Ferrari, A. J., 1978. Mars topography harmonics and
geophysical implications, J. geophys. Res., 83, 3497-3508.
Bills, B. G., Kieffer, W. S. & Jones, R. L., 1987. Venus gravity: A
harmonic analysis, J. geophys. Res., 92, 10 335-10351.
Blitzer, L., Boughton, E. M., Kang, G. & Page, R. M., 1962.
Effect of ellipticity of the equator on 24-hour nearly circular
satellite orbits, J. geophys. Res., 67, 329-335.
Chao, B. F., 1989. Comment on ‘Moment of inertia of
three-dimensional models of the Earth’ by T. Tanimoto,
Geophys. Res. Lett., 16, 1075.
Chao, B. F. & Gross, R. S., 1987. Changes in the Earth’s rotation
and low-degree gravitational field induced by earthquakes,
Geophys. J.R. astr. SOC., 91, 569-596.
Chao, B. F. & Rubincam, D . P., 1989. The gravitational field of
Phobos, Geophys. Res. Left., 16, 859-862.
Chao, B. F. & Au, A. Y., 1991. Temporal variation of the Earth’s
low-degree gravitational field caused by atmospheric mass
redistribution: 1980-1988, J. geophys. Res., 96, 6569-6575.
Dickey, J. 0. & Schutz, R. E., 1989. Earth rotation and references
studies, EOS, Trans. A m . geophys. Un., 43, 1062.
Goldreich, P. & Toomre, A , , 1969. Some remarks on polar
wandering, J. geophys. Res., 74, 2555-2567.
Heiskanen, W. A,, 1962. Is the Earth a triaxial ellipsoid?, J.
geophys. Res., 67, 321-327.
Heiskanen, W. A. & Moritz, H., 1967. Physical Geodesy, Freeman,
San Francisco.
Hubbard, W. B., 1984. Planetary Inferiors, Van Nostrand,
Reinhold, New York.
Izsak, I. G., 1961. A determination of the ellipticity of the Earth’s
equator from the motion of two satellites, Asfr. J., 66,
226-229.
Jackson, J. D., 1975. Classical Electrodynamics, 2nd edn, Wiley,
New York.
Jeffreys, H., 1976. The Earth, 6th edn, Cambridge University Press,
Cambridge.
Kaula, W. M., 1979. The moment of inertia of Mars, Geophys. Res.
Lett., 6, 194-196.
Kinoshita, H., 1977. Theory of the rotation of the rigid Earth,
Celest. Mech., 15, 277-326.
Kohnlein, W., 1966. The geometric structure of the Earth’s
gravitational field, in Geodetic Parameters for the 1966
Smifhsonian Institution Standard Earth, vol. 111, eds Lundquist ,
C. A. & Veis, G., SAO Spec. Rep. 200, Cambridge, MA.
Downloaded from http://gji.oxfordjournals.org/ at Pennsylvania State University on May 17, 2016
precisely to the great geoid low in the Indian Ocean, as
might be expected. Note that, for the real, non-rigid Earth,
any change in p(r) due to mass redistribution will cause (6)
and (8) to change. Chao & Gross (1987) have studied the
effect of earthquakes, while Chao & Aug (1990) studied for
the atmospheric fluctuations; the changes are found to be
too small to detect.
Chao & Gross (1987, p. 585) have presented the correct
relations (5a,b) but inadvertently left out the factor 1/2 in
(5b) in figuring the a axis orientation as quoted there. Chao
(1989) made the same error. This bears on Tanimoto’s
(1989) model predictions based on heterogeneous Earth
models obtained from seismic tomography: Tanimoto’s
predictions, as a result, agree somewhat less satisfactorily
with the observations than claimed.
One can express the equator in terms of an equivalent
geoid ‘ellipse’-equivalent in the sense of its rotational
dynamics (see below). Equation (8) gives the orientation of
the semi-major axis of this ellipse, while the difference in
the semi-major axis and the semi-minor axis is given by
fl(Cq,
S:2)1’2R= 69.4 m. The latter, obtained by using
Bruns formula on equation (l), agrees with that found by
Kohnlein (1966; see also Baker & Makemson 1967; Torge
1980). This ellipse, however, does not correspond to a
‘best-fitting’ geometrical figure for the equator, simply
because there are many other Stokes coefficients that also
contribute significantly to the equatorial geoid figure. In
fact, it is known that the (sea-level) difference between the
western Pacific Ocean geoid high (near the a axis) and the
Indian Ocean geoid low (along the b axis) reaches some
180 m.
A similar phenomenon occurs on Mars but on a much
grander scale, dominated by the great Tharsis rise:
C,, = -84.79 x lop6, S,, = 48.64 x
(Balmino, Moynot
& Vales 1982). Thus, Martian a axis lies along the (75.loE,
104.9”W) diameter [this corrects an error in Bills & Ferrari
(1978) which mislabelled the a and b axes due to the n / 2
ambiguity mentioned above; Bills 1990, personal communication]. It points to the centre of the Tharsis
topographic high. Martin (B - A)/MR2 is about 2.52 X
corresponding to a difference in the geoid semi-major
and semi-minor axes of almost 1.3 km. Its (B - A ) / ( C - A )
ratio is as large as about 118. The Moon, Venus and Phobos
are the only other planetary bodies for which the said
parameters have been determined or inferred (see Kopal
1966; Bills et al. 1987; and Chao & Rubincam 1989,
respectively). Unlike the Earth (but Goldreich &
Toomre 1969) and Mars, these slowly rotating bodies are
truly triaxial bodies in the sense that the quantities C - A ,
C - B, and B - A are of the same order of magnitude.
The Earth’s y (equation 7), although small, is not without
dynamical consequences. The axial torque exerted by the
Sun and the Moon due to the non-zero y generates
variations in the Earth’s rotational speed at semi-diurnal
tidal periods. The maximum peak-to-peak amplitude
amounts to some 0.075 milliarcsecond in (longitudinal)
orientation, or 0.005ms in the Universal Time (UT1) over
every 12hr (Liu & Chao 1990). Since the signal-to-noise
ratio (in the frequency domain) of coherent harmonic
signals increases as the square root of the number of
observations, these semi-diurnal signals may become
detectable in the near future as more accurate observations
701
702
H . S. Liu and B. F. Chao
Kopal, Z., 1966. An Introduction to the Study of the Moon, Reidel,
New York.
Liu, H. S. & Chao, B. F., 1990. Semidiurnal longitudinal libration
in the Earth’s rotation, EOS, Trans. Am. Geophys. Un., 71,
1271.
Lorell, J., 1972. Estimation of gravity field harmonics in the
presence of spin-axis direction error using radio tracking data,
J . Astronaut. Sci., XX, 44-54.
Marsh, J. G. et al., 1990. The GEM-T2 gravitational model, J.
geophys. Res., 95, 22 043-22 071.
Morse, P. M. & Feshbach, H., 1953. MefhodF of Theoretical
Physics, McGraw-Hill, New York.
Scheidegger, A. E., 1982. Principles of Geodynamics, SpringerVerlag, New York.
Tanimoto, T., 1989. Reply to Chao, Geophys. Res. Letf., 16,
1076-1077.
Torge, W., 1980. Geodesy, Walter de Gruyter, New York.
Wagner, C. A., 1973. Does AZ2 vary?, J. geophys. Rex, 78,
470-475.
Woolard, A. E., 1953. Theory of the rotation of the Earth around
its center of mass, Astr. Pap. Am. Ephemeris XV, 1.
Yoder, C. F., Williams, J. G. & Parke, M. E., 1981. Tidal
variations of Earth rotation, J. geophys. Res., 86, 881-891.
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