GEOPHYSICAL RESEARCH LETrERS, VOL. 18, NO. 11, PAGES 2007-2010, NOVEMBER 1991
LIBRATION
IN THE
EARTH'S
ROTATION
B. F. Chao,
• D. N. Dong,
2H. S.Liu,• andT. A. Herring
2
Abstract. External luni-solar torque exerted on the
difference(B-A) of the Earth's two equatorialprincipal
momentsof inertia givesrise to two typesof librational
inertial space.In thispaperwe usethe angularmeasureof
milliarcsecond
(1 mas = 1/15 ms in UT1). We concentrate on the spin and polar motion; the corresponding
motions
nutation can be readily obtained from, e.g., the Poinsot
"cone"construction[cf. Sasaoand Wahr, 1981].
Qualitatively, the libration can be understood as
follows. Let the three principal momentsof inertia of the
triaxial Earth beA <B < C, and the corresponding
principal
in the Earth's
rotation:
the semidiurnal
libration
in spinand the progradediurnallibrationin polarmotion.
Formulasfor the librations consideringa realistic Earth
model and their tidal decompositionsare derived and
evaluated.The spinlibrationhasa maximumpeak-to-peak
amplitude of 0.09 milliarcseconds,that of the polar
libration is 0.06 milliarcseconds.
Implicationsconcerning
their detectabilityand role in the tidal variation of Earth
axes be a, b, and c. Precession and nutation are the
advances,especially those made possible by intensive
observingcampaigns[Clark et al., 1990; Herring, 1990],
have set the stage for high-frequencyEarth rotation
researchwhichhasnot beenreadilyaccessible
in the past.
This new arenais rich in geophysical
signals,mostnotably
the diurnal and semidiurnalvariationsof tidal origin.One
consequenceof tidal torques exerted on the Earth's
equatorial bulge embodiedin the quantityC-A. By the
same token, the libration is causedby the tidal torque
exertedonB-A. Simplistically,
underdiurnalspinthe tidal
torque on B-A will acceleratethe spin when the a axis
approaches
the Earth-Sun/Moon line, and deceleratethe
spin after the a axis passesthat line. The result is a
semidiurnallongitudinal "spin libration", superimposed
uponthe otherwiseuniformdiurnalspin.If we saythat the
Earth precesseslike a top, it librates like a physical
pendulum. In addition, in the presenceof a non-zero
obliquity(the angulardepartureof the equator from the
ecliptic),the tidal torquewill alsohavenon-zerocomponents in the equatorial plane, producing a latitudinal
"polarlibration"which manifestsitself as a diurnal polar
motionin the progradedirection(counterclockwise
asseen
from abovethe north pole in the terrestrialframe). Since
B-A is onlyabout1/150 of C-A [Liu and Chao,1991],we
expectthe libration to be small.
The Eulerian theory of the rotational dynamicsof a
rigid, triaxial Earth in the luni-solargravitationalfield has
been treated by, e.g.,Routh [1891],and later by Woolard
[1953]and Kinoshita[1977].The presentpaper adoptsa
direct and simple approachwhich is more akin to modern
representation of the Earth's gravitational field. This
approach has traditionally been used in treating the
ellipsoidal Earth in precession/nutationstudies [e.g.,
of them is the libration.
Melchior, 1983]. We shall also allow for an Earth model
rotation
are discussed.
Introduction
The Earth's rotation varies slightlywith time. These
variationsare of two distinctdynamictypes:thosecaused
by geophysicalprocesses(massredistributionand mass
movement) under the conservationof total angular
momentum,and thoseinduceddirectlyby externallunisolartidal torquesthat alter the angularmomentum.Wellknown examples of the latter include tidal braking,
precessionand nutation.The presentpaperwill studyyet
another variation due to the direct tidal torque, called
libration [Liu and Chao, 1990] becauseof its generic
similaritywith the familiar libration of the Moon.
Keepingpacewith advancesin spacegeodeticdetermination of Earth orientationin both accuracyand frequency, geophysicalresearchon Earth rotation dynamicshas
been
marked
with
The Earth's
an ever-refined
3-dimensional
time
rotational
scale.
Recent
variation
can be
conveniently separated into two components:the 1dimensionalvariation in the spin and the 2-dimensional
variation
in
the
rotational
axis
orientation.
Libration
affects both variations. The spin is often expressedin
terms of the length-of-day,but for sub-dailyvariations
under study here the more appropriatemeasureis the
Universal Time (UT1). The axis orientationvariation is
calledgenericallythe polar motion as seenin the terrestrial reference
frame
and the nutation
when viewed
from
that is more realisticin two ways:an elasticallydeformable mantle enclosinga fluid core. We shall see that the
elastic deformation in particular makes a significant
differencefor the polar libration. We make two further
assumptions
aboutthe core:it contributesnothingto the
observedstaticB-A and doesnot activelyparticipatein
the libration.The former assumptionleavesthe observed
B-A all to the mantle;howtrue it is in realityis presently
unknown, hence any observationof the librations can
providecriticalinformationon thisassumption.
The latter
assumptionseemsplausibleconsideringthe rapid nature
of the libration.
XGeodynamics
Branch,
Goddard
Space
FlightCenter
2Dept.Earth,Atmosphere
andPlanetary
Sciences,
General Formulationfor the Torque
MassachusettsInstitute of Technology
Considera body of densitydistributiono(r') and total
Copyright1991by the AmericanGeophysical
Union.
mass M'.
Papernumber91GL02491
0094-8534/91/91GL-02491
For
convenience
we choose our coordinate
originat the centerof massof the body.Let the external
perturbingpoint massM be at positionr. Then the total
$03.00
2007
2008
Chao et al.: Libration in Earth Rotation
torque exerted on the bodyM' is
Spin Libration
L = GM[pR-2r'x•dV'
(1)
where G is the gravitational constant, R = r - r', A
denotesunit vector,and the integrationis over the volume
The first term in equation(4a) involvingC2• and S21
would generatea diurnal variation in the spin.Theoretically, C2• and S2•vanishidenticallyif the terrestrialz axis
is chosento alwayscoincidewith the c axis,andnominally
V' of the body. Substitutingr' with r-R, equation (1)
all the Stokes coefficients
reduces
to L = rx[GMpR-2fldV
'. The integralin this
system.In practice such a choice may not be feasible,
especiallyin the presenceof short-periodpolar motion.
But failing to do that only causessecondorder effectson
the other Stokes coefficients.Here we shall simply
disregardthis term.
The secondterm involvingC22and S22is the driving
torque for our spin libration.Liu and Chao [1991] have
demonstrated
that C22andS22are relatedto the equatorial
principalmomentsof inertiaA and B by
expressionhappensto be the total gravitationalforce on
the body M' due to M, which equalsthe negativeof the
total force on M, at r, due to M'. The latter can be
written as -MVU(r), so that
L=
-MrxVU(r)
(2)
where U is the externalgravitationalpotentialof M'.
Now let the body M' be our triaxial Earth with
C>B>A. Approximatethe externalperturbingbody,Sun
or Moon, by the point massM at positionr = (r, o, A) in
the geocentricsphericalcoordinates,so that o and A are
the co-latitudeand longitudeof the sub-solaror sub-lunar
point on the Earth. o undergoeslong-periodvariations,
whereasA movesin the retrogradesensediurnally. U is
customarilyexpressedin termsof the harmonicexpansion
[e.g.,Kaula, 1966]:
e(l')
=GMt
E•
r
ß
1--0 rn--0
cosmA+
refer
to such a coordinate
C22+ iS22
= (B - A) e•aø/ (4 M' a02)
(5)
where A0 is the longitudeof the a axis.Combiningequation (4a) with (5) onegetsthe torquefor the spinlibration
Lz = 3GM(B
-A)sin2o
sin2(A
-Ao). (6)
2r 3
This expression
dulyreflectsthe tidal origin(dependence
nature(theangular
dependence
of
Plm(COSO)
(3) ofr-3),thesemidiurnal
sinmA)
whereP/mis the associated
Legendrefunctionof degreel
andorderm. The (un-normalized)Stokescoefficients
Clm
and S/mare evaluatedwith referenceto some radius a0
enclosingthe body,normallytaken to be the mean radius
of the Earth.
The degree/=0 term representsthe monopole,whose
gradientlies alongr, yieldinga zero torque.The degree
/=1 terms vanish identically given that the coordinate
origincoincides
with the centerof mass.So the non-trivial
terms start with degree /=2. In fact, they are the only
terms that need be retained here because,as r>a0 in the
case of the Earth-Sun or Earth-Moon system,the higher
degreetermsdiminishrapidlywith l. Substituting
equation
(3) into (2) and convertingit into Cartesianxyz coordinates, we get
L z = Lo[ sin2o(C21sinA
- S21cosh)
+ 4sin20(C22sin2A
- S22cos2A)
], and
(4a)
;f_,= Lo[ iC2osin20eia
2A), as well as the dependenceon B-A and the angle
A-A 0 between the a axisand Earth-Sun/Moon line.
For a deformablebody,B -A will be modifiedunder the
applicationof L• and this feedsback to equation(6). It
canbe argued,however,that thisdeformationis negligible
so far as spin libration is concerned[Dongand Herring,
1990].The equationof motionfor the spinis
Cma'h =
(7)
where • is the mean spinrate. Here the useof the polar
momentof inertia of the mantleCm• 0.89Creflectsthe
exclusionof the fluid corein the libration,and, following
Munk and MacDonaM [1960], the Earth's (variable)
rotation
velocity
vectoris givenby to-- (mx,my,l+m•)rl,
with mi • 1.
Substituting
equation(6) into (7) andintegratingtwice
yields the spin libration as a function of time. As an
approximation,by holdingthe orbital parametersr and o
fixedand onlylettingA varydiurnallyin the integration,we
get the spin libration
• UT(t)
- 83I2GM
B-A sin20
sin2(A
-Ao). (8)
2r3 Cm
- i/2(C21+iS21)(1
+ 3cos2o)
+ i(C21-iS21)sin20
e2i'x
- 2/ (C224-iS22)
sin2oe-ia]
The factor(B-A)/Cm= 2.48x 10-5, asevaluated
from
(4b)
equation (5) using the observedC22and S22[Liu and
Chao, 1991].The maximumhalf-amplitude(at 0 = 90 ø
where
Lo= (3/2)GMM'
ao2/r
3,and[, --Lx+ iLy(~denotes when the Sun or Moon "cross"the equator)is 0.014 mas
due to the Sun, and 0.031 mas due to the Moon.
complex variable). The combined luni-solar torque is
simplythe sumof the torquesfrom the Sun and the Moon
For a more detailedand accurateevaluationof the spin
individually.
Lz perturbs
the spinwhereas/5,
excites
the libration • UT one can simplyinvoke the tidal expansion
polar motion.
for the time variationsin r and 0 in equation(8). We
Chao et al.: Libration
demonstratethis in the Appendix.Yoderet al. [1981]have
givena simplifiedexpression
for the principallunar and
solarterms,but unfortunatelytheirvaluesweremistakenly
magnifiedby a factorof 2•r.It sufficesto sayherethat the
maximum,combinedluni-solar,semidiurnal,peak-to-peak
amplitudeof the spinlibrationis about0.09 mas.
in Earth Rotation
2009
sophisticated
treatment(directlyon equation9) hasbeen
givenby Dongand Herring[1990].Here we onlypoint out
that the maximum,combinedluni-solar,diurnal, peak-topeakamplitudeof the polarlibrationcanreachabout0.06
mas.
Discussion
Polar
Libration
We have seen that the spin and polar librations have
Now examine equation (4b). Note first that since A
maximum,peak-to-peak
amplitudes
on the orderof <0.1
moves
diurnally
in theretrograde
sense,
thefactore/A(or mas.PreciseVLBI determinationsof Earth rotationduring
arenowbeginning
to approach
the0.1
e-iA)represents
a retrograde
(or prograde)
diurnalpolar intensivecampaigns
motion;
andsimilarly
foret2• withrespect
to semidiurnal mas resolution for sub-daily determinations[J. Ray,
polar motion.
The first term in (4b) corresponds
to the retrograde
diurnal polar motion associated
with the precessionand
personalcommunication,
1991].Considerfurtherthe fact
nutation:
giventhatC2o= -(C-A)/M• aoz,it isequivalent
also varies due to tidal currents in the ocean and tidal
to, e.g.,equation(2.39) ofMelchior[1983].The secondand
third termsinvolvingCz•and$z•will againbe disregarded
here.Theyrepresentan apparent,complexmotionwhich
wouldvanishuponthe selectionof the terrestrialz axisto
deformationsin the solidEarth and ocean[e.g.,Yoderet
al., 1981;Broscheet al., 1989].Thesetidal variationshave
The lastterm in (4b) is the drivingtorquefor our polar
libration.Substitutingequation(5) into it yields
4r 3
tidal
periods.Independent
of thelibration,theEarth'srotation
amplitudes
typically
severaltimeslargerthanthelibration,
but with the same tidal periods.
The implication,then,is two-fold.In orderto studythe
coincide with the c axis.
I2= - 3iGM(B-A)
sin20
e-i(a-2ao) .
that the librations are of diurnal and semidiurnal
(9)
Sasaoand Wahr[1981]havegiventhe generalequationof
motion for the "observable"
nutationand the corresponding polar motionbasedon the 1066AEarth Model and a
non-participating
fluid core. Here, for illustration,let us
first present the simple elastic "whole-Earth"case as
treated in Munk and MacDonald[1960]:The equationof
motion
isrh = ia(fit-•) where
fit -- mx+imy.
Thefree
Chandlerfrequencya can be written as (C-A)fl/(kA),
wherek= 1.43 is the Chandlertransferfunction.The polar
motion
excitation
function
9 isgiven
byik,4L/[a2A(C
-A)].
Combiningtheseandholdingr and0 fixedyields,approxi-
mately,
fit = -3GM(B-A)(4fl2r3A
)-1sin20e-i(a-aO
). This
is essentially
the sameresultobtainedbyKinoshita[1977]
for a rigidEarth. Note that the dependence
of A reflects
the progradediurnal nature of the motion.The correspondingnutationis progradesemidiurnal.
Now usingSasaoand Wahr's[1981]formulation,it can
bereadilyshown
fromtheirequations
(3.20,3_.21)
thatthe
effectof the Earth's deformationinducedby L at prograde
diurnalperiod is to reducethe abovefit amplitudeby a
factorof 0.480 [Dongand Herring,1990].This reduction,
although severe, presumablyreflects the reality more
closelyat the shortperiodsin question.Thus, the polar
libration
fit(t)= - 0.36
G.•_M
BA-A sin20
e-i(A-2A0)
. Cl0)
fl 2r3
Its maximumhalf-amplitude
(absolutemagnitudeof fit) is
0.0089masdue to the Sun (at 0=66.5 ø), and 0.022mas
dueto the Moon (at 0=61.5ø). Again,a tidal decomposition of equation(10) can be invokedvia tidal expansions
for r and o, as presentedin the Appendix. A more
diurnal
and semidiurnal
tidal
variations
in the Earth's
rotation, it is necessaryto recognizeand allow for the
presenceof the librations.Our studyprovidesa set of
predictedlibration valuesfor this purpose.On the other
hand, the fact that the libration amplitude is barely
discernible in individual measurements does not mean it
is undetectable
in time records. This is because the
libration, as any tidal oscillation,is coherentin time and
hence enjoysthe ,/N "processing
gain"in the signal-tonoise ratio, where N is the number of observationsin the
time record. Thus, a reasonablylong time record can
readilyreveal the signalsin a Fourier spectralanalysis.
Furthermore,longtimerecordsare necessary
for resolving
neighboringtidal components
in spectralanalyses.Typically,a recordof a few months(with sufficientsampling
rate, of course)shouldbe sufficientin analyzingthe major
componentsin conjunctionwith the tidal signals.
Finallylet usexaminethe sameeffecton otherplanets.
The libration,of course,is the dominanttidal forcingfor
secondaryplanetarybodiesorbitinga primarybody,such
as the Moon relative to the Earth, Phobosto Mars, and
Mercuryto the Sun.For suchbodies,key constraints
on
the parameter(B-A)/C are providedby librationamplitude observations.
Mars has spinperiod and obliquitysimilarto thoseof
the Earth, but its (B-A)/C is 30 timeslarger[cf.Liu and
Chao, 1991] and the Sun is 1.5 timesmore distant.The
resultant libration, due te the Sun, is thus about 10 times
larger than that on the Earth. This is still a rather small
angle;sonot surprisingly
the availableViking determinationsof Martian rotationare far from adequateto detect
it [cf.Reasenberg
andKing,1979].
On theotherhand,although
muchsmallerin (B-A)/C
(7.9x 10-6)[cf.Mottinger
etal., 1985],Venushasa much
slowerrotation(225Earth days,retrograde)anda shorter
distanceto the Sun (0.72 AU). The net effectis a spin
libration
thatis4 x 104timeslargerthanthatoftheEarth
dueto the Sun,at a semidiurnal
periodof 112 days.This
amountsto one arcsecondin the maximumpeak-to-peak
amplitude,equivalentto 30 m of surfacedisplacement.
2010
Chao et al.: Libration in Earth Rotation
The corresponding
amplitudefor the polarlibrationis
severaltimessmallerthanthat(dueprimarilyto thesmall
by VLBI, EOS Trans.Amer. Geophys.Union,71, 1271,
1990.
obliquity
of 177ø), witha retrograde
diurnal
periodof 225 Dong,D. N., and T. A. Herring,Observedvariationsof
days.The Magellanspacecraft
hasan orbitnavigation UT1 and polar motion in diurnal and semidiurnal
bands,EOS Trans.Amer. Geophys.Union,71,482, 1990.
uncertainty
of about+100 m, and the Venussurface
Herring,
T. A., Geodeticresultsfrom the October1989
resolution
of the on-boardsynthetic
apertureradaris also
ExtendedR&D VLBI experiment,EOS Trans.Amen
on the order of +100 m. It remains to be seen whether
Geophys.Union,71, 1271, 1990.
Magellanobservations,
whenintegrated
overthelifetime
of the mission(nominally
a few rotationcycles),
canbe
usefulin detectingVenus'librations.
Kaula, W. M., Theory of Satellite Geodesy,Blaisdell,
Massachusetts,1966.
Kinoshita,H., Theory of the rotationof the rigid Earth,
Appendix:Tidal Expansionfor Librations
Cel. Mech., 15, 277-326, 1977.
Liu, H. S., and B. F. Chao, Semi-diurnal longitudinal
Approximatetidal decompositionfor the libration can be
invoked via tidal expansionof the time variation in r
(Earth-Sunor Earth-Moondistance)ando (co-latitudeof
sub-solaror sub-lunar point on Earth) following the
developmentin Chapter I of Melchior [1983]. These
expressionsare then substitutedinto equation (8) to
evaluate spin libration and into equation(10) for polar
libration, leadingto Table A1 belowwhichgivesthe halfamplitudesfor the few major terms.
Table A1. Major Tidal Amplitudesof Earth's Libration.
Tidal Term
Period (hr)
Amplitude(mas)
M2
S2
N2
K2
SemidiurnalSpin Libration
(L. Princ.)
12.420
0.029
(S. Princ.)
12.000
0.013
(L. Elliptic)
12.662
0.0048
(S/L Declin.) 11.968
0.0036
K1
O1
P1
Q1
M1
J1
ProgradeDiurnal Polar Libration
(S/L Declin.) 23.935
0.015
(L. Princ.)
25.819
0.010
(S. Princ.)
24.066
0.0047
(L. Elliptic)
26.868
0.0008
(L. Elliptic)
24.833
0.0008
(L. Elliptic)
23.099
0.0008
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D. N. Dong and T. A. Herring, EAPS, Massachusetts
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02139.
(ReceivedSeptember6, 1991;
acceptedSeptember26, 1991.)