ARTICLES
The chaoticobliquityof the planets
J. Laskar& P. Robutel
Astronomie
et SystdmesDynamiques,
Bureaudes Longitudes,TT
AvenueDenfert-Rochereau,
F-75014Paris,France
Itlumericalstudy of the global stability of the spin-axisorientation(obliquity)of the planetsagainst
secularolbital pedurbationsshows that all of the tefiestrial planetscluld haveexperiencedlarge,
chaotic variationg in obliquity at aome time in the past, The obliquity of Mars is still in a large
chaoticregion,langing fiom 0o to 60o. Mercuryand Venushavebeenstabilizedby tidal dissipation,
andthe Ea h may havebeenstabilazedby captureof the Moon.Noneof the obliquitiesof the terreskial
planetscan ther6forebe consideredas primordial.
Tue problem of the origin of the obliquities of the planets (that
is, the orientation of their spin axis) is important, becauseif the
obliquities are primordial, they could provide some dynamical
constraints on the formation of the Solar Systeml-4.We have
investigatedhow long-term perturbations by other planets affect
the obliquities and precession rates of all major planets of the
Solar System, and demonstrate that none of the inner planets
(Mercury, Venus, the Earth and Mars) can be considered to
have primordial obliquities. Any of these planets could have
started with nearly zero obliquity, in a prograde state, and the
chaotic behaviour of their obliquity could have driven them to
their current value. If their primordial spin rate was high enough,
Mercury and Venus could have undergone large and chaotic
changes of obliquity from 0o to -90o, before dissipative effects
ultimately drove them to their current obliquity. The present
obliquity of the Earth may have been reached during a chaotic
state before the capture of the Moons. The obliquity of Mars is
chaotic at present, with possible variations from 0oto *60o. The
obliquities of the outer planets are stable, but may have gone
through a similar processin an earlier stageof the Solar System's
formation.
Methods
If the motion of the Solar Systemwere quasiperiodic,the motion
of precession could be modelled by equations (3) and (5) in
Box 1. One can then use the argument of resonanceoverlap6to
suggestwhether the motion of precessionwill become chaotic,
or will stay regular. This can be done for the outer planets, but
to obtain more accurate results, we integrated directly the full
equations of precession (equations (3) and (4)), and did a
frequency analysis (Box 2) of the precession solution. As the
dynamical ellipticity (C - A)/C is proportionalTto v2, the pre-
cessionconstant a is proportional to the spin rate of the planet
z. For a given value of a, we numerically integrated the precessionequations, for all values of the initial obliquity, in steps
of 0.1'. These integrations use the orbital solution La90 (ref. 8)
as an input and are, in general, done over 18 Myr with a step
size of -200 yr. We then used frequency analysis to obtain a
frequency curve whose regularity tells us about the regularity
of the precessionand obliquity. We also recorded the maximum
and minimum value reached by the obliquity during these
integrations (Figs 3, 4).
The primordial spin rates of Mercury, Venus and the Earth
are not known precisely; so for these planets, we did the above
analysis for a wide range of a to obtain a general view of how
the obliquity changedwith time (Fig. 5). In Fig. 5, regular motion
is representedby small dots, while the chaotic zones are larger
black points. The chaotic zones would be even larger if we took
into account the chaotic diffusion of the orbital solution over
timescalesof a billion yQars8.
Past and present obliquities
Mercury. Mercury's present spin is very low, and apparently
trapped in a (2:3) spin-orbit resonance. Most probably, the
Frequency(" yr-1)
-40
-20
0
20
40
-40
-20
0
20
40
-b
-8
-9
-6
-7
-8
-9
FlG.1 Fundamentalplanes for the definition of precession.Eg, and Ec, are
the mean equator and ecliptic at time t Eco is the fixed ecliptic at Julian
date 2000, with equinox 7o. The general precessionin longitude ry'is defined
by ,y':A -O. y'o is definedby y'oN:yoN:O; i* is the inclination.
608
FlG.2 Fourierspectrum (with Hanningfilter) of the planetaryforcing term
in inclinationA(t) + iB(t). The logarithm of the amptitude of the Fourier
coefficients is plotted against the frequency (" yr-1).
. VOL361 . 18 FEBRUARY
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primordiai spin rate of Mercury was much higher and was
slowed down by solar tides. Extrapolatingthe apparentcorrelation of angular momentum with mass', the primordial rotation
period of Mercury is found to be 19 h (ref. 10), indicating a
t. Tidal dissipationand coreprecessionfrequencyof -l27" yr
mantle interaciionswill causeihit ftequencyto decreasett'tt't'.
For a rotation period smaller than 100h, there still existssome
regular motion for small obliquities.,but when the spin rate of
for a :20" yr-', the obliquity entersa very
the planet decreases,
large ch ao tic zon e, e x t ending f r om 0o t o - 100" ( Figs 3 , 5 4 ) .
During this period, and until tidal dissipationultimately drives
Mercury into its presentstate,the orientation of Mercury undergoes very large and chaotic changes from 0o to 100" within a
i.* miliion years. For example, for a : 10"yr-t (Fig' 3),
the obliquity varies from 0 to more than 90" within a few
million years for any value of the initial obliquity within this
range.
Providedits primordial rotation period was smallerthan 300 h
(Fig. 5), Mercury must therefore have suffered large-scale
chaotic behaviour during its history. At some point its orientation must have been very different from at present (with the
pole facing the Sun) and this could have left some traces on
the planet.
BOX 1
Precession and obliquitY
Let A = B < C be the principalmoments of inertiaof a planet.we assume
that the axis of rotation of the planet is also its axis of maximum
mom e n t u mo f i n e r t i a C T h e sp in a n g u la rm o m e n tu mis H:HH, w i th H a
unit vector, and H = Km9l2v: Cy, where K is a coefficientdependingon
the internalstructureof the planel(K:2/5 for an homogeneoussphere),
/? its equatorialradius,rn its mass, and y its spin rate. Let ? be the unit
vector in the direction of the sun, and r the distance to the sun. The
torque exerted by the Sun, limited to first order in {t/r,is
(1)
*{ i*n
G the gravitationalconstant, and M
where / is the matrix of inertia23'2o,
=
the solar mass. The precessionmotion of the planet is givenby dH/dt l.
By averagingover the time (and thus over the mean anomaly),we obtain
the equationsof precession;that is, the secularmotion of the spin axis
of the olanet
dH
r l i = a ( 1 - e 2 \- 3 /2 ( A.2 ) xx?
dr
2 a "v
c* n
C
with a the semi-major axis of the planet, will be called the precession
constant.Let E"o be the orbital plane at the origin of time with equinox
(Fig. 1), and N the
1'.o,E"t the orbital plane at time t, with equinox y
inlersectionof E"oand 8",. The precessionin longitudeis 4':,1 - O. where
O = yoN is the longitudeof the node,and A:7N. We also let 7[ be the
The coordinatesof H in the reference
point on F", such that 7iN:o.
frame E", with origin y6 are (sin ry'sin s, cos r, sin e, cos r), where s is
(F",)
the obliquity(the angle between the Equator(€o,)and orbital plane
at time t). After some calculus,and using the action variable x:cos e
*/2 ) co s ( O) w here i * i s
and t h e n o t a t i o n sP = sin ( l*/2 ) sin ( O) , q : sin ( i
the equations of
we
obtain
E"o,
to
the inclinationof 8", with respect
precession
a7(
dX
d7(
dr AX
or
o4t
d{t
(3)
which are related to the hamiltonian
g(u, a, t)=9(t - e(ilIae'2x2
2
+'/t-
NATURE. VOL 361
x ' t e( t ) s in, i,+ 8( t ) c osr / )
1993
18 FEBRUARY
w i th
A (t):2@ + p\qp - P i l \/'[r-
P '- q'
B \tl :2( p - q(qp pqD /,[7 - p' 4
C (t)=(qi - pq)
The expressi onA (t)+18(t) and the eccentri ci tye(t) descri beth e orbi tal
motion of the planet and will be given by the LagO solution of Laskars.
As the orbital motion of the planets,and especiallyof the inner planets,
25 27 a quasi peri odiapproxi
of A (t) + l 8(f ) i s not w el l
c
mati on
i s chaoti c818
,
suited for obtainingan accuratesolutionover a few millionye?rs.This is
reflectedin the f ilteredFourierspectrumof A(t) + i8( t) for the innerplanets
(Fig.2) where only the main peaks can be identifiedas combinationsof
the planetarysecularfundamentalfrequencies.A quasiperiodicapproximati on of A (t)+18(t) of the form
N
Q)
where 2 is tne unit vector in the directionof orbital angularmomentum,
e the eccentricity,and where
o :3 G!l
Venus. The orientation of the pole of Venus (178') is one of
Dissipativeeffects
the puzzlingfeaturesof the SolarSysteml'r3'1o.
of core-mantle interactionsand atmospherictides can account
for some change in spin orientation, but apparently only from
90' to the present valuel3'ta.It is thus generally assumedthat
the retrograderotation of Venus is primordial, which is one of
the argumentssupportingthe stochasticacretionmechanismfor
the terrestrialplanets3'4.
As for Mercury, we investigatedthe global dynamics of the
precessionand obliquity of Venus using frequency analysis.
According to Goldreichto, the primordial rotation period of
Venus *ui -13 h, which givesa precessionconstantof 31" yr-l.
We started our investigation with a frequency constant of
40"yr-1, which revealsa regularbehaviourfor small obliquities,
and then a large chaoticregion from 50'to 90'. Tidal dissipation
due to the Sun will decreasethe planet's rotation speed, and
consequentlyits precessionconstant.When the precessioncont , c o r r e s p o n d i n gt o a p e r i o d o f - 20 h , th e
s t a n t r e a c h e s2 0 " y r
c h a o t i c z o n e e x t e n d sf r o m 0 o t o n e a r l y 9 0 " ( F i g . 3 ) . In Fi g .3 ,
we can seeseveralwell definedchaotic zones,but the frequency
analysisshows that no invariant surfacebounds the motion, so
the orbits can dilluse in a few million years from 0' to nearly
90'. As the precessionfrequencycontinuesto decrease,the shape
(4)
,q( t) + a( t) :
I
an u;( r ;t+d" t
k- l
can still be used when lookingat the outer planets or, more important,
for a qualitativeunderstandingof the behaviourof the solution.With this
the hami l toni anreads
approxi mati on.
u :!0_
2
xI
e( r)2) ttz yz * i l
cusi n(zr.f+Q+6)
* yz
(5)
which is the hamiltonianof an oscillatorof frequencyaX, perturbedby a
quasiperiodicexternaloscillationwith severalfrequenciesz*, Resonance
cos s rs equal to the opposite (-v") of one
wifl occur when ry'=aX:a
of the frequencies zr. When limited to a single periodic term, and with
e(t):Cte, this hamiltonian is integrable and is often referred to as
Colombo'stop28.lts equilibriumpoints are then called Cassini states2s.
But in the present case,e(t) and A(t) + l8(t) take into accountthe secular
perturbationsof the whole Solar System,modelledas a system with 15
degreesof freedom,and this hamiltonianhas 1+15 degreesof freedom.
Its sotutionsevolve in a phase space of drmension2 +30 (2 accountsfor
the rr. X variables).In fact, the orbital solutionLa9O is not coupledwith
the orecessionvariables,and will thus never change.The hamiltonianis
thus consideredto dependon time, but in a nonperiodicway. Traditional
tools, such as the Poincar6 surface of sections, cannot be used for the
analysisof its dynamics,and we will rely on Laskar'smethod of frequency
(Box 2).
analvsiss30'31
609
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of the chaoticregion changes(Fig. 5). Ifthe obliquity of Venus
reaches-90', it could becomestabilizedaroundthat valueand
evadethe chaotic zone. Dissipativeeftectscould then, as the
planet rotation continuesto slow down, drive the spin axis
tourardsits present value of 178". Therefore the rehograd€
rotation of Venus may not be primordial. If the orientation
underwentlarge-scalechaotic behaviourduring its history,the
pole of Venuscould have faced the Sun for extendedperiods,
possiblycausingstrongchangesin the climateand atmospheric
it much.Wardrsshowedthat the planetarysecularperturbations
causethe obliquity of MaN, unlike that of the Earth, to sufter
large variations of 110" around its mean value 25". The precesiionconstantof Ma$16J7(8.26'yr t) is closeto somesecular
frcqucnciesof the planet, and the questionof resonancehas
beenraisedseveraltimesrs'r?.
Without knowtedgeofthe precise
initial conditions,no definiteconclusionhas been drawn.
H€re we inv€stigatethe p.oblem in a difierent manner,by
looking at its global dynamics. First we did the frequercy
circulation.
analysisof the obliquity of Mars over 45 Myr. We found a large
Mars, The analysisfor Mars is more dilect, asits rotationspeed
can be consideredas primordial. Mars is far from th€ Sun and
hasno largesatellitewhosetidal interactionscould haveslowed
chaotic zone ranging from -0' to 60'. In Fig. 4a and 4 the
motion looks regular for small values of the initial obliquity.
But the motion ofthe inner planetsis chaotic3'r3,
and the phases
of the planetarysccularterms are lost after -100 Myr, We can
\,^^,,^
,'-_,)^,,.,.-1\
v'"ur \u-4w
v'
yr-r)
Mercury(u,=10"
thus study the problem independently of the phases.The results
in Fig. 4c, d were calculated with a small change of the initial
10
10
rc
WX 2 Frcquencyanalysig
The numericalanalysisof the fundamentalfrequenciesfollows the
methodof ref. 8. In that case,frequencyanalysisshowedthat for tfre
innerplanets(Mercuryto Mars)the chaoticzoneis relativelylarge,
whereasfor the outerplanets(Jupiterto Nepture)this zoneis much
smaller.Moregenerally,
the methodoutlirredherecan be appliedto
studythe stabilityof the solutionsof a conservative
dynamicalsystem,
andis basedon a refinednumerical
searchfor a quasiperiodic
approximationof its solutionsovera finitetimespan8s31.tt f(t) is a function
with valuesin the complexdomainobtair€dnumerically
over a finite
time span I-f,f|, the frequencyanalysisalgorithmwill consistof a
searchfor a quasiperiodic
approximation
for f(t)with a finitenumber
of periodicterms of the form
a_n
i
(U
E 10 0
c
(g
()
E
50
c
q)
N
f(t)100
50
eo(degrees)
0
1 00
50
€o(degrees)
FlG. 3 Frequencyanalysis of the obliquity over 18 Myr for Mercury (a b)
with o :!O" yr-t (rotationperiod -255 h) and Venus(C d) with a:2O" y(-7
(rotation period -20 h). The precessionfrequency is plotted against the
initial obliquity.The regularityof the frequencycurve shows the regularity
of the motion. For Mercury,the chaotic zone extends from 0'to 100", and
from 0o to nearly 90'for Venus.The maximum,mean and minimumvalues
of the obliquity reached over 18 Myr are shown in b and d.
10
;4{qi
tr
T'
0
o...,
. ,\
.
X
(d
E- roo
100
c
G
c)
E.
.=
h {l
c
;
Thefrequencies
o,, andcomplexamplitudesak arefoundby iteration.
To determinethe first frequency01, one searchesfor the maximum
anplitudeof f(6)=(f(t),e'"'), wherethe scalarproductof two functions(f(t),g(fD is definedby
(f(t),e(r))= f(t)8(t)x(f)dt
*l_,
andwhereX(f) is a weightingfunction,that is, a positivefunctionwith
t/27 I -r x(t) dt=1, andg the complexconjugateof g. Oncethe first
periodicterm e-rr is found,its complexamplitudea, is obtainedby
projection,
orthogonal
andtheprocessis startedagainontheremaining
part of the functionf1(t)=f(t)-a1e'"". A secondanalysisis made
to estimate the precisionof the determination.In the case of an
integrablehamiltoniansystem with n &grees of freedom,the
frequencyanalysisof the solutionswitl give their quasiperiodic
expansionand in particularwill determin€the vector(r,),:t,, of the
fundamentalfrequenciesof the system.
lf an orbit is not regular(not quasiperiodic),
in case of nearly
integrablesystems,the frequerrcyanalysisgives a quasiperiodic
approximation
to the solutionwhichholdsonlylocallyin time.In other
words,it will give us a frequencyvector(zi(f))i for eachvalueof 1
obtainedby applyingthe frequencyanalysisalgorithmover the time
span [1 f + f]. For the analysisof the precessionand obliquity,we
shallonly look for the precessionfrequencyp, whichis associatedto
te angleand actionvariables,r, X.The initial phaserlo is fixed.lf we
take someinitialconditionsX we cancarryout the frequencyanalysis
for the orbitscorresponding
to initialconditionsWo,X) (at t:0) over
the time span [O,f]. We thus @fine the frequencymap
Fri R-R
0
0
50
100
0
50
100
so(degrees)
FlG.4 Frequencyanalysis of the obliquity of Mars over 45 Myr. In a and b,
the planetarysolution is used with the present initial conditions.A large
chaotic zone is.visible, ranging from 0" to 60'. When the phase of the
planetary solution is shifted by about -3 Myr, this large chaotic zone is
even more visible (c, d). The maximum,mean and minimum values of the
obliquityreachedover 45 Myr are shown in b and d.
610
T 8p elnxt
k-1
X +F
The regularityof the precessionand obliquitycan be analysedvery
preciselyby studyingtf€ regularityof the frequencymap of this
problemwith 1+n degreesof freedom.The distortionsof the
frequencymap indicatethe non-existence
of invariantsurfaes which
mayboundthe obliquityvariationss'3l.As thesedistortionsincrease,
theyproducea completeloss of regularityof the frequencymap,which
is an indicationof chaoticmotion31.
. VOL361 . 18 FEBRUARY
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date of the orbital solution (--3 Myr relative to Fig.4a, b).
This should convince the reader that the chaotic zone extends
from nearly 0" to 60'. Figure 4d shows that some orbits changed
dramatically in less than 45 Myr.
From this, we can conclude that the obliquity of the planet
Mars is chaotic, with possible variations ranging from nearly 0'
to -60". We also computed the Liapounov exponent of the
obliquity of Mars for the presentinitial conditionsreand found
a value simila r to th at of t he planet r 8,- ll5 M y r . W e c h e c k e d
that this value reveals the intrinsic chaotic behaviour of the
obliquity of Mars by repeating the computation with a 'regular'
value (50"yr-l) of precessionconstant.Nevertheless,we consider the frequency analysis as more significant, as it gives the
extent of the chaotic zone. Moreover, by giving the global picture
of the dynamics, it shows that small changes in the initial
conditions or model will not affect this result substantially.
The chaotic behaviour of the orientations of Mars could have
been important in its evolution, as its obliquity could have gone
40h
60h
1 0 0h
20h
200h
40h
300 h
B 0h
r
x
15 h
8h
20h
24h
12h
20h
24h
48h
48h
Obliquity(deg)
FlG.5 The zone of large-scalechaotic behaviourfor the obliquity of (a)
Mercury,(b) Venus,(c) tne Earth (without the Moon),(d) Mars, for a wide
range of spin rate. The precession constant a is given on the left in
arcsecondsper year,and the estimate of the correspondingrotationperiod
of the planet on the right, in hours. The regularsolutions are represented
by small dots, while large black dots represent solutions with large-scale
chaotic behaviour.The chaotic motion is estimated by the diffusionof the
precession frequency obtained from numerical frequency analysis over
36 Myr. For each initial condition(e, a), the correspondingpoint is shifted
vertically proportionallyto the diffusion of the orbit. In the large chaotic
. VOL361 . 18 FEBRUARY
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1993
zonesvi si bl ehere,the chaoti cdi ffusi onw i l l occuron hori zontall i nes (a i s
fixed),and the obliquityof the planet can explore horizontallyall the black
zone in a few millionyears.The extent of the chaoticzone should be even
largerwhen one considersdiffusionover longertimescales.With the Moon,
one can considerthat the present situationof the Earthcan be represented
i n c roughl yby the poi nt e:23", a:55"yr 1, w hi ch i s i n the m i ddl eof a
large zone of regular motion. Without the Moon,for a spin period ranging
from about 12 h to 48 h, the obliquityof the Earth would suffer very large
chaoticvariations.
ARTICLES
up to 60" during its history. The obliquity of Mars cannot be
considered as primordial; the planet also could have.been formed with any obliquity from 0o to 60o, and been driven to its
present values only by the eftect of the secular planetary perturbations.
The Ealth. The equations for the precession of the Earth are
complicated by the presence of the Moon; we consider the
stability of its obliquity in the accompanyingpapert. The present
obliquity variations2oare only *1.3'around the mean value of
23.3", but in the absence of the Moon, the torque exerted on
the Earth would be smaller; we found, assuming a primordial
rotation of 15 h, that the obliquity of the Earth would show very
large variations in a chaotic zone ranging from nearly 0o to 85o
(ref. 5).
The value of the primordial spin rate is controversial, so we
decided to study the global stability of the obliquity for a wide
range of rotation period (Fig. 5c). For a rotation period of 8 h,
the increase of the equatorial bulge of the Earth would compensate for the lack of the Moon2r, but for more generally
acceptedvalues of its spin, ranging from 12 to 48 h, most initial
values.of obliquity lead to large-scalechaotic variations, in many
casesreaching 80-90'(Fig. 5).
For the Earth, as for the other terrestrial planets, the obliquity
cannot be considered primordial. The Earth could have been
formed in a large chaotic zone, and the capture of the Moon
would have frozen the obliquity to its current value.
The outer planets. We also analysed the global stability of the
obliquity of the outer planets (Jupiter, Saturn, LJranus,
Neptune). The orbital solution of these planets behaves in a
very difterent way from those for the inner planets. As the
difterent spectra of A( r) + tB( t) show (Fig. 2), they consist only
of well isolated lines. There will be no large overlap of the
different resonant terms as for the inner planets, and the solutions are essentiallyreg.rlar. Estimatesof precessionconstants
for the outer planets are imprecise becausetheir structure is not
well known, but are estimated22to be well below 5" y.-t. As we
obtain a global picture of the dynamics, we also learn how the
obliquities behave for slightly difterent values of theseconstants.
Practically speaking, it is difficult to destroy the stability of the
obliquities until the precessionconstant of theseplanets reaches
26" yr-r,where it will be in resonancewith the secularfrequency
Received26 October1992; accepted25 January1993.
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6L2
su (ref. 8). Even then, this resonancebeing isolated, the motion
will show large but regular oscillations. We can thus say that,
as for the orbital motion, the obliquities of the outer planets
are essentiallystable, and should be considered as primordial.
But primordial here means that the obliquities have not
changed since the Solar System was in the final state of its
formation, in a state similar to its present one (at least for the
outer planets). If it had, at some previous stage, been more
massivethan now, similar instabilities as for the inner planets
could have occurred, which may have driven the obliquities of
the outer planets to their present configurations.
Gonclusions
None of the inner planets (Mercury, Venus,the Earth and Mars)
can be considered as to have primordial obliquities, and all
theseplanets could have been formed with a near-zeroobliquity.
The obliquities of all these planets could have undergone largescalechaotic behaviour during their history. Mercury and Venus
have been stabilized by dissipative eftects, the Earth may have
been stabilized by the capture of the Moon, and Mars is still in
a large chaotic zone, ranging from 0o to 60'.
It should also be noted that planets in a retrogradestatewould
be much more stable than planets with a prograde spin rate, as
all the main forcing frequencies of the inclination are negatives
(Fie. 2).
The obliquities of the outer planets (Jupiter, Saturn, Uranus
and Neptune) are essentiallystable,and can thus be considered
as primordial, that is, with about the same value they had at
the end of the formation of the Solar System. Nevertheless,
chaotic behaviour of the obliquities under planetary perturbations could have occurred in an earlier stage of the formation
of the Solar System, when it could have been more massive,
and this point should be clarified by further studies.
The chaotic orbital motion of the inner planets8'rSresults in
large-scalechaotic behaviour of their obliquity for most initial
conditions. The stability of the Earth's climate thus depends on
the presence of the Moon which stabilizes its obliquitys, and
hence the insolation variations on its surface. This should be
taken into consideration when estimating the probability of
finding a planet with climate stability comparable to the Earth's
around a nearby star.
n
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ACKNOWLEDGEMENTS.
We thank P. Barge, A. Chenciner, S. Dumas, D. Gautier, M. Herman,
F. Joutel, P. Y. Longaretti, S. Peale and B. Sicardy for discussions, and J. Marchand for technical
assistance.
. VOL361 . 18 FEBRUARY
1993
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