TEMPORAL VARIATIONS OF ROTATIONAL ORIGIN
IN THE ABSOLUTE VALUE OF GRAVITY
KURT LAMBECK
Research Group o f Space Geodesy, Paris Observatory*)
Summary:
When correcting precise gravity measurements for polar motion, the Earth's rotational deformation must be considered, as this will increase the correction based on a rigid Earth by
about 15%. Conversely the gravity observations can be used to estimate the Love numbers h 2 and k 2
at the Chandler frequency.
In a recent paper Bur~a [1] discussed the temporal variations in the absolute value
of gravity at any point on the Earth's surface due to the Earth's variable rotation.
In his treatment he considered the Earth to be essentially a rotating rigid body but
it is necessary to allow for the Earth's rotational deformation as well. The gravity
measurements would then provide an additional means of measuring the Earth's
elastic response to forces of intermediate duration.
The potential of the centrifugal force at a point distant 6 from the instantaneous
rotation axis is:
A U c = ½(D26 2 .
I f coi denote the components of the Earth's rotation about a set of body fixed axes xi,
then to 2 = ~co 2 and the direction cosines of the rotation axis are ah/co. Also:
i
(~2 = ~ x 2
i
_ (Z(.oixi/(D)2
i
r2 = ~x2
i
and AU c can be rearranged as:
(1)
AUg =
z
,o=r +
2
+
2
- 2x 2) - 6Z~oio~i+lXiXi+t}
i
=
i
= ½o~2r2 + AU~.
The first term is a radial component and the second term represents second degree
zonal, tesseral and sectorial spherical harmonics. I f we denote the mean rotation rate
by 12; coi = 12rn i for i = 1, 2, 093 = t2(1 + rna) and co2 ~ O2(1 + 2rn3)if we neglect
terms of the order f22rn 2. The rn 1 and m z represent with adequate accuracy the angular coordinates of the rotation axis in the x~ system and m 3 represents the variations
in the rotation rate, or
m 3 = -a(l.o.d)/(1.o.d.),
where 1.o.d. denotes length of day. The m 1 and m 2 are of the order 10 -6 and the m s
is of the order 10 -a.
*) A d d r e s s : 9 2 - M e u d o n ,
studia geoph, et geod. 17 (1973)
France.
269
K. Lambeck
The radial term in (1) will cause a radial deformation u r if the Earth is compressible
and as the total mass is conserved there is a variation in g at the surface of the Earth
by an amount
g/g = --2ur/r .
If we assume that the Earth is homogeneous we have at r -- R [7; p. 143]
Ur =
Q2m 3
1
15X+#
,
+2#
where 2 and # are the Lam6 constants. If we assume 2 = # = 1.5 x 1012 dynes/era z,
we obtain:
~g
8 R202m3
-
g
-
75
~, 10 - 1 2
#
The radial term of (1) can henceforth be ignored.
The only possibly significant terms in the harmonic part of (1) are:
AU'~ -- lf22(1 + 2m3)(x 2 + x 2 - 2x 2) - ~2X3(mlX 1 -t" rrt2x2)
and the time varying part is
~ U ; = ½~22 rrt3(t ) (x 2 q- X 2 -- 2x 2) -- f22x3(rrt1(t) x 1 -}- m2(t ) x2) ----
= rn3QZrZ( c°s2 go - 2) - QZrZ sin go cos go Ago,
where Ago is the "variation in latitude" of the station.
5U'c(t) is the time dependent part of the potential of the centrifugal force and will
cause redistribution of mass inside the Earth. This is the same redistribution that
causes the elongation of the period of the Earth's Eulerian nutation from a theoretical
value of 10 months for a rigid Earth to an observed value of about 14 months, and
is best described by the Love numbers h2, k2 V6]. The potential of the deformation
is k2 5U'c and the radial deformation of the Earth's surface is h2 5U'c/g. The change
in gravity 9 at the Earth's surface then is [3]:
(2)
Ag(t) = - 2 ( 1 + h2 - 3k2) (~U'c(t)
R -1
=
= f22r2R -1 (1 + h 2 - -3zk2)[sin 2go Ago - 2m a (cos 2 go - 2 ) ] ,
where R is the Earth's radius.
The first term represents the influence on g of the Earth's wobble and will exhibit
two principle components, one of 14 months (the free nutation or Chandler period)
and a forced annual period that is mainly of meteorological origin. The second term
in Ag(t) represents the variations in g due to the variable rate of rotation about the
instantaneous axis.
For the Love number k2 in the above expression (2) the value derived from the
Chandler wobble would appear to be the appropriate choice and using Jeffrey's [4]
270
Studia geoph. Ct geod. 17 (1973)
Temporal Variations of Rotational Ori#in...
period one obtains k 2 = 0.294. This is a dynamic L o v e n u m b e r in the sense that it
includes the effect o f the dynamic response of the liquid core to the Earth's rotation
[8]. N o comparable observational value exists for the Love n u m b e r hz but a probable
value is 0.60 _ 0-05. With R = 6.8 x 10 s cm, f2 = 7.3 x 10 -5 tad/s, Arp = 10 -6,
the double amplitude o f the first term of (1) is about 8 ~tgal for mid-latitudes. F o r
rn 3 = 10 -8 the double amplitude of the second term is about 0-2 ggal.
Recent measurements o f absolute gravity m a d e by Sakuma [9] are accurate to
about 1 ggal and they a r e therefore sensitive to the variations in the distance to the
Earth's instantaneous rotation axis and a continuous series o f observations would
give a value of the elastic factor (1 + h2 - ~k2) for the periods 12 and 14 months.
The latter is p r o b a b l y of greater interest as seasonal variations o f tidal, meteorological
a n d hydrological origin will also occur in gravity and will be difficult to separate.
Alternativel3~, if we assume that h e and k2 are k n o w n we could use the gravity
measurements to determine the polar motion as suggested by Burga [1] but We would
require a way of modelling accurately the other above mentioned variations in gravity.
The effect o f the variable rate o f rotation is t o o small to be detected with the present
instrumentation, but the rapid changes in m 3 that m a y reach 10 -8 in a time interval
o f as little as 5 days [2] and t h a t are of meteorological origin [5] could be detected
in the future by the precise relative gravimeters that are n o w being developed [10].
Reviewer: M. Burda
Received 6. 3. 1973
References
[1] M. Bur ga: Variations of the Earth's Gravity Field due to the Free Nutation. Studia geoph.
et geod., 16 (1972), 122.
[2] B. G u i n o t : Work of the Bureau International de l'Heure on the Rotation of the Earth.
In Earthquake Displacements and the Rotation of the Earth, Ed. L. Mansinha, D. E. Smylie
and A. E. Beck, D. Reidel Publ. Comp., Dordrecht 1970.
[3] H. Jeffreys: The Earth, 4th edition, Cambridge Univ. Press, Cambridge 1962.
[4] H. Jeffreys: Variation of Latitude. Monthly Notices, Royal Ast. Soc., 141, (1968), 54.
[5] K. L a m b e c k , A. C a z e n a v e : The Earth's Rotation and Atmospheric Circulation, II, the
continuum. Submitted to the Geophys. Journ. Royal Astr. Society, 1973.
[6] A. E. H. Love: The Yielding of the Earth to Distribution Forces. Proc. Roy. Soc., London,
Ser. A., 82 (1909), 255.
[7] A. E. H. L o re: A Treatise on the Mathematical Theory of Elasticity. 4th edition, Cambridge
Univ. Press, Cambridge 1927.
[8] W. H. Munk, G. J. F. M a c D o n a l d : The Rotation of the Earth. Cambridge Univ. Press,
Cambridge 1960.
[9] A. S a k u m a : Observations experimentales de la constance de la pesanteur au Bureau International des Poids et Mesures -- S~vres. Bull. Gdod,, N ° 100 (1971), 159.
[10] W. A. P r o t h e r o , J. M. G o o d k i n d : Earth Tide Measurements with the Superconducting
Gravimeter. J. Geophys. Res., 77 (1972), 926.
studia geoph, et geod. 17 (1973)
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