ARTICLES The chaoticobliquity of the planets J. Laskar& P. Robutel Astronomieet SystdmesDynamiques, ,TT Avenue Denfert-Rochereau, F-75014 , France

Itlumericalstudy of the global stability of the spin-axisorientation (obliquity) of the planetsagainst secularolbital pedurbationsshows that all of the tefiestrial planetscluld haveexperienced large, chaotic variationg in obliquity at aome time in the past, The obliquity of is still in a large chaoticregion, langing fiom 0o to 60o. Mercuryand Venushave been stabilized by tidal dissipation, andthe Ea h may havebeen stabilazed by captureof the Moon.None of the obliquitiesof the terreskial planetscan ther6forebe consideredas primordial.

Tue problem of the origin of the obliquities of the planets (that cessionconstant a is proportional to the spin rate of the planet is, the orientation of their spin axis) is important, becauseif the z. For a given value of a, we numerically integrated the pre- obliquities are primordial, they could provide some dynamical cessionequations, for all values of the initial obliquity, in steps constraints on the formation of the Solar Systeml-4.We have of 0.1'. These integrations use the orbital solution La90 (ref. 8) investigatedhow long-term perturbations by other planets affect as an input and are, in general, done over 18 Myr with a step the obliquities and precessionrates of all major planets of the size of -200 yr. We then used frequency analysis to obtain a Solar System, and demonstrate that none of the inner planets frequency curve whose regularity tells us about the regularity (, , the and Mars) can be considered to of the precessionand obliquity. We also recorded the maximum have primordial obliquities. Any of these planets could have and minimum value reached by the obliquity during these started with nearly zero obliquity, in a prograde state, and the integrations (Figs 3, 4). chaotic behaviour of their obliquity could have driven them to The primordial spin rates of Mercury, Venus and the Earth their current value. If their primordial spin rate was high enough, are not known precisely;so for these planets, we did the above Mercury and Venus could have undergone large and chaotic analysis for a wide range of a to obtain a general view of how changesof obliquity from 0o to -90o, before dissipative effects the obliquity changedwith time (Fig. 5). In Fig. 5, regular motion ultimately drove them to their current obliquity. The present is representedby small dots, while the chaotic zones are larger obliquity of the Earth may have been reached during a chaotic black points. The chaotic zones would be even larger if we took state before the capture of the Moons. The obliquity of Mars is into account the chaotic diffusion of the orbital solution over chaotic at present,with possible variations from 0oto *60o. The timescalesof a billion yQars8. obliquities of the outer planets are stable, but may have gone through a similar processin an earlier stageof the Solar System's Past and present obliquities formation. Mercury. Mercury's present spin is very low, and apparently trapped in a (2:3) spin-orbit resonance. Most probably, Methods the 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 resonanceoverlap6 to Frequency(" yr-1) suggestwhether the motion of precessionwill become chaotic, -40 -20 0 20 40 -40 -20 0 20 40 or will stay regular. This can be done for the outer planets, but to obtain more accurate results, we integrated directly the full -b 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- -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 FlG.2 Fourierspectrum (with Hanningfilter) of the planetaryforcing term date 2000, with equinox 7o. The general precessionin longitude ry'is defined in inclinationA(t) + iB(t). The logarithm of the amptitude of the Fourier by ,y':A -O. y'o is definedby y'oN:yoN:O; i* is the inclination. coefficients is plotted against the frequency (" yr-1). 608 NATURE. VOL361 . 18 FEBRUARY1993 ARTICLES

(178') primordiai spin rate of Mercury was much higher and was Venus. The orientation of the pole of Venus is one of effects slowed down by solartides. Extrapolating the apparentcorrela- the puzzlingfeatures of the SolarSysteml'r3'1o. Dissipative tides can account tion of angular momentum with mass',the primordial rotation of core-mantleinteractions and atmospheric from period of Mercury is found to be 19h (ref. 10), indicating a for some changein spin orientation,but apparentlyonly t. precessionfrequency of -l27" yr Tidal dissipationand core- 90' to the presentvaluel3'ta. It is thus generallyassumed that is one of mantle interaciionswill causeihit ftequencyto decreasett'tt't'. the retrograderotation of Venus is primordial, which for For a rotation period smallerthan 100h, there still existssome the argumentssupporting the stochasticacretion mechanism regular motion for small obliquities.,but when the spin rate of the terrestrialplanets3'4. of the the planet decreases,for a :20" yr-', the obliquity entersa very As for Mercury, we investigatedthe global dynamics large chaotic zone, extending from 0o to -100" (Figs 3,54). precessionand obliquity of Venus using frequency analysis. period of During this period, and until tidal dissipationultimately drives According to Goldreichto, the primordial rotation of 31"yr-l. Mercury into its presentstate, the orientation of Mercury under- Venus*ui -13 h, which givesa precessionconstant goes very large and chaotic changes from 0o to 100" within a We started our investigation with a frequency constant of i.* miliion years. For example, for a : 10"yr-t (Fig' 3), 40"yr-1, which revealsa regularbehaviour for small obliquities, the obliquity varies from 0 to more than 90" within a few and then a largechaotic region from 50'to 90'. Tidal dissipation million years for any value of the initial obliquity within this due to the Sun will decreasethe planet's rotation speed,and consequentlyits precessionconstant. When the precessioncon- range. t, Providedits primordial rotation period wassmaller than 300h stant reaches20"yr correspondingto a period of -20h, the (Fig. 5), Mercury must therefore have suffered large-scale chaotic zone extendsfrom 0o to nearly 90" (Fig.3). In Fig.3, chaotic behaviour during its history. At some point its orienta- we can seeseveral well definedchaotic zones,but the frequency motion, so tion must have been very different from at present (with the analysisshows that no invariant surfacebounds the nearly pole facing the Sun) and this could have left some traces on the orbits can dilluse in a few million years from 0' to the shape the planet. 90'. As the precessionfrequency continues to decrease,

BOX 1 Precession and obliquitY

Let A = B < C be the principalmoments of inertiaof a planet.we assume with that the axis of rotation of the planet is also its axis of maximum - q' momentumof inertiaC The spin angularmomentum is H:HH, with H a A(t):2@ + p\qp Pil\/'[r- P'- H = Km9l2v: Cy,where K is a coefficientdepending on unit vector, and B\tl:2( p - q(qp pqD/,[7- p' 4 the internalstructure of the planel(K:2/5 for an homogeneoussphere), /? its equatorialradius, rn its mass, and y its spin rate. Let ? be the unit C(t)=(qi - pq) vector in the direction of the sun, and r the distance to the sun. The the orbital torque exerted by the Sun,limited to first order in {t/r,is The expressionA(t)+18(t) and the eccentricitye(t) describe motion of the planet and will be given by the LagOsolution of Laskars. (1) As the orbital motion of the planets,and especiallyof the inner planets, *{i*n 25 27 is chaotic818 , a quasiperiodicapproximation of A(t) + l8(f) is not well an accuratesolution over a few millionye?rs. This is gravitational and M suited for obtaining where / is the matrix of inertia23'2o,G the constant, innerplanets = reflectedin the f ilteredFourier spectrum of A(t) + i8( t) for the mass. The precessionmotion of the planetis givenby dH/dt l. the solar (Fig.2) where only the main peaks can be identifiedas combinationsof By averagingover the time (and thus over the mean anomaly),we obtain the planetarysecular fundamental frequencies. A quasiperiodicapproxima- equationsof precession;that is, the secularmotion of the spin axis the tion of A(t)+18(t) of the form of the olanet

N dH rli=a(1 -e2\-3/2(A.2)xx? Q) ,q(t) + a( t) : I an u;(r;t+d"t dr k-l

where 2 is tne unit vector in the directionof orbital angularmomentum, can still be used when lookingat the outer planets or, more important, e the eccentricity,and where for a qualitativeunderstanding of the behaviourof the solution.With this c * n approximation.the hamiltonianreads o:3G!l 2a"v C ttz u :!0_ e(r)2) yz * il * yz with a the semi-major axis of the planet, will be called the precession 2 constant.Let E"o be the orbital plane at the originof time with equinox (Fig. 1'.o,E"t the orbital plane at time t, with equinox y 1), and N the xI cusin(zr.f+Q+6) (5) inlersectionof E"oand 8",.The precession in longitudeis 4':,1 - O. where O = yoN is the longitudeof the node,and A:7N. We also let 7[ be the which is the hamiltonianof an oscillatorof frequencyaX, perturbedby a point on F", such that 7iN:o. The coordinatesof H in the reference quasiperiodicexternal oscillation with severalfrequencies z*, Resonance frame E", with origin y6 are (sin ry'sin s, cos r, sin e, cos r), where s is (F",) wifl occur when ry'=aX:a cos s rs equal to the opposite (-v") of one the obliquity(the angle between the Equator(€o,) and orbital plane of the frequencies zr. When limited to a single periodicterm, and with at time t). After some calculus,and using the action variablex:cos e */2) * e(t):Cte, this hamiltonian is integrable and is often referred to as and the notationsP = sin(l*/2) sin (O), q: sin(i cos (O) where i is Colombo'stop28. lts equilibriumpoints are then called Cassinistates2s. the inclinationof 8", with respect to E"o, we obtain the equations of But in the presentcase, e(t) and A(t) + l8(t) take into accountthe secular precession perturbationsof the whole Solar System,modelled as a system with 15 freedom. d{t a7( dX d7( (3) degreesof freedom,and this hamiltonianhas 1+15 degreesof in phase spaceof drmension2 +30 (2 accountsfor dr AX or o4t Its sotutionsevolve a the rr. X variables).In fact, the orbital solutionLa9O is not coupledwith which are related to the hamiltonian the orecessionvariables, and will thus never change.The hamiltonianis thus consideredto dependon time, but in a nonperiodicway. Traditional g(u, a, t)=9(t - e(ilIae'2x2 tools, such as the Poincar6surface of sections, cannot be used for the 2 analysisof its dynamics,and we will rely on Laskar'smethod of frequency +'/t- x'te(t)sin,i, +8(t)cos r/) (4) analvsiss30'31(Box 2).

609 NATURE. VOL361 18 FEBRUARY1993 ARTICLES of the chaoticregion changes(Fig. 5). Ifthe obliquity of Venus it much.Wardrs showed that the planetarysecular perturbations reaches-90', it could becomestabilized around that valueand causethe obliquity of MaN, unlike that of the Earth, to sufter evadethe chaotic zone. Dissipativeeftects could then, as the large variations of 110" around its mean value 25". The pre- planet rotation continuesto slow down, drive the spin axis cesiionconstant of Ma$16J7(8.26'yr t) is closeto somesecular tourardsits present value of 178". Therefore the rehograd€ frcqucnciesof the planet, and the questionof resonancehas rotation of Venus may not be primordial. If the orientation beenraised several timesrs'r?. Without knowtedgeofthe precise underwentlarge-scale chaotic behaviour during its history,the initial conditions,no definiteconclusion has been drawn. pole of Venuscould have faced the Sun for extendedperiods, H€re we inv€stigatethe p.oblem in a difierent manner,by possiblycausing strong changes in the climateand atmospheric 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 chaotic zone ranging from -0' to 60'. In Fig. 4a and 4 the can be consideredas primordial. Mars is far from th€ Sun and motion looks regular for small valuesof the initial obliquity. hasno largesatellite whose tidal interactionscould haveslowed But the motion ofthe inner planetsis chaotic3'r3,and the phases of the planetarysccular terms are lost after -100 Myr, We can thus study the problem independently of the phases.The results (u,=10"yr-r) \,^^,,^v'"ur ,'-_,)^,,.,.-1\ Mercury \u-4wv' in Fig. 4c, d were calculated with a small change of the initial 10

rc 10 WX 2 Frcquencyanalysig a_n The numericalanalysis of the fundamentalfrequencies follows the methodof ref. 8. In that case,frequency analysis showed that for tfre innerplanets (Mercury to Mars)the chaoticzone is relativelylarge, whereasfor the outerplanets (Jupiter to Nepture)this zoneis much smaller.More generally, the methodoutlirred here can be appliedto studythe stabilityof the solutionsof a conservativedynamical system, i andis basedon a refinednumerical search for a quasiperiodic (U approxi- E 100 mationof its solutionsover a finitetime span8s31. tt f(t) is a function c with valuesin the complexdomain obtair€d numerically ()(g overa finite E time spanI-f,f|, the frequencyanalysis algorithm will consistof a 50 searchfor a quasiperiodicapproximation for f(t)with a finitenumber c of periodicterms of the form q)

N f(t) T 8p elnxt - k-1 50 100 0 50 100 eo(degrees) €o(degrees) Thefrequencies o,, andcomplex amplitudes ak arefound by iteration. To determinethe first frequency01, one searchesfor the maximum FlG.3 Frequencyanalysis of the obliquity over 18 Myr for Mercury (a b) anplitudeof f(6)=(f(t),e'"'), wherethe scalarproduct of two func- (f(t),g(fD with o :!O" yr-t (rotationperiod -255 h) andVenus (C d) with a:2O" y(-7 tions is definedby (rotation period -20 h). The precessionfrequency is plotted against the initial obliquity.The regularityof the frequencycurve shows the regularity (f(t),e(r))= f(t)8(t)x(f)dt of the motion.For Mercury,the chaoticzone extendsfrom 0'to 100", and *l_, from 0o to nearly90'for Venus.The maximum,mean and minimumvalues andwhere X(f) is a weightingfunction, that is,a positivefunction with of the obliquity reached over 18 Myr are shown in b and d. t/27 I -r x(t) dt=1, andg the complexconjugate of g. Oncethe first periodicterm e-rr is found,its complexamplitude a, is obtainedby orthogonalprojection, and the process is startedagain on the remaining partof the functionf1(t)=f(t)-a1e'"". A secondanalysis is made 10 to estimate the precisionof the determination.In the case of an integrablehamiltonian system with n &grees of freedom,the ;4{qi frequencyanalysis of the solutionswitl give their quasiperiodic tr T' expansionand in particularwill determin€the vector(r,),:t,, of the fundamentalfrequencies of the system. lf an orbit is not regular(not quasiperiodic),in case of nearly 0 integrablesystems, the frequerrcyanalysis gives a quasiperiodic approximationto the solutionwhich o..., holdsonly locally in time.In other . ,\ . words,it will give us a frequencyvector (zi(f))i for eachvalue of 1 obtainedby applyingthe frequencyanalysis algorithm over the time X span [1 f + f]. For the analysisof the precessionand obliquity,we (d E-roo 100 shallonly look for the precessionfrequency p, whichis associatedto c G te angleand actionvariables ,r, X.The initial phaserlo is fixed.lf we c) E. take someinitial conditions X we cancarry out the frequencyanalysis for the (at .= h{l orbitscorresponding to initialconditions Wo,X) t:0) over c the time span [O,f]. We thus @fine the frequencymap ; Fri R-R 0 0 50 100 0 50 100 X+F so(degrees) The regularityof the precessionand obliquitycan be analysedvery preciselyby studyingtf€ regularityof the frequencymap of this FlG.4 Frequencyanalysis of the obliquity of Mars over 45 Myr. In a and b, problemwith 1+n degreesof freedom.The distortionsof the the planetarysolution is used with the present initial conditions.A large frequencymap indicatethe non-existenceof invariantsurfaes which chaotic zone is.visible, ranging from 0" to 60'. When the phase of the maybound the obliquityvariationss'3l. As thesedistortions increase, planetarysolution is shifted by about -3 Myr, this large chaotic zone is theyproduce a completeloss of regularityof the frequencymap, which even more visible (c,d). The maximum,mean and minimum values of the is an indicationof chaoticmotion31. obliquityreached over 45 Myr are shown in b and d. 610 NATURE. VOL361 . 18 FEBRUARY1993 ARTICLES date of the orbital solution (--3 Myr relative to Fig.4a, b). that this value reveals the intrinsic chaotic behaviour of the This should convince the reader that the chaotic zone extends obliquity of Mars by repeatingthe computation with a 'regular' from nearly 0" to 60'. Figure 4d shows that some orbits changed value (50"yr-l) of precessionconstant. Nevertheless, we con- dramaticallyin lessthan 45 Myr. sider the frequency analysis as more significant, as it gives the From this, we can conclude that the obliquity of the planet extent of the chaotic zone. Moreover, by giving the global picture Mars is chaotic, with possible variations ranging from nearly 0' of the dynamics, it shows that small changes in the initial to -60". We also computed the Liapounov exponent of the conditions or model will not affectthis result substantially. obliquity of Mars for the presentinitial conditionsreand found The chaotic behaviour of the orientations of Mars could have a value similar to that of the planetr8,-ll5 Myr. We checked been important in its evolution,as its obliquity could have gone

40h

60h

100h 20h

200h 40h 300 h B0h

r

x

15 h 8h

20h 24h 12h 20h

48h 24h 48h

Obliquity(deg)

FlG.5 The zone of large-scalechaotic behaviourfor the obliquityof (a) zonesvisible here, the chaoticdiffusion will occuron horizontallines (a is Mercury,(b) Venus,(c) tne Earth (without the Moon),(d) Mars,for a wide fixed),and the obliquityof the planet can explorehorizontally all the black range of spin rate. The precession constant a is given on the left in zone in a few millionyears. The extent of the chaoticzone shouldbe even arcsecondsper year,and the estimate of the correspondingrotation period largerwhen one considersdiffusion over longertimescales. With the Moon, of the planet on the right, in hours.The regularsolutions are represented one can considerthat the presentsituation of the Earthcan be represented by small dots, while large black dots representsolutions with large-scale in c roughlyby the point e:23", a:55"yr 1, whichis in the middleof a chaoticbehaviour. The chaotic motion is estimated by the diffusionof the large zone of regularmotion. Without the Moon,for a spin period ranging precession frequency obtained from numerical frequency analysis over from about 12 h to 48 h, the obliquityof the Earthwould suffer very large 36 Myr. For each initial condition(e, a), the correspondingpoint is shifted chaoticvariations. verticallyproportionally to the diffusion of the orbit. In the large chaotic NATURE. VOL361 . 18 FEBRUARY1993 ARTICLES up to 60" during its history. The obliquity of Mars cannot be su (ref. 8). Even then, this resonancebeing isolated, the motion considered as primordial; the planet also could have.been for- will show large but regular oscillations. We can thus say that, med with any obliquity from 0o to 60o, and been driven to its as for the orbital motion, the obliquities of the outer planets present values only by the eftect of the secular planetary per- are essentiallystable, and should be considered as primordial. turbations. But primordial here means that the obliquities have not The Ealth. The equations for the precession of the Earth are changed since the Solar System was in the final state of its complicated by the presence of the Moon; we consider the formation, in a state similar to its present one (at least for the stability of its obliquity in the accompanyingpapert. The present outer planets). If it had, at some previous stage, been more obliquity variations2oare only *1.3'around the mean value of massivethan now, similar instabilities as for the inner planets 23.3", but in the absence of the Moon, the torque exerted on could have occurred, which may have driven the obliquities of the Earth would be smaller; we found, assuming a primordial the outer planets to their present configurations. 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 Gonclusions (ref. 5). None of the inner planets (Mercury, Venus,the Earth and Mars) The value of the primordial spin rate is controversial, so we can be considered as to have primordial obliquities, and all decided to study the global stability of the obliquity for a wide theseplanets could havebeen formed with a near-zeroobliquity. range of rotation period (Fig. 5c). For a rotation period of 8 h, The obliquities of all these planets could have undergone large- the increase of the equatorial bulge of the Earth would com- scalechaotic behaviour during their history. Mercury and Venus pensate for the lack of the Moon2r, but for more generally have been stabilized by dissipative eftects, the Earth may have acceptedvalues of its spin, ranging from 12 to 48 h, most initial been stabilized by the capture of the Moon, and Mars is still in values.of obliquity lead to large-scalechaotic variations,in many a large chaotic zone, ranging from 0o to 60'. casesreaching 80-90'(Fig. 5). It should also be noted that planetsin a retrogradestate would For the Earth, as for the other terrestrial planets, the obliquity be much more stable than planets with a prograde spin rate, as cannot be considered primordial. The Earth could have been all the main forcing frequencies of the inclination are negatives formed in a large chaotic zone, and the capture of the Moon (Fie. 2). would have frozen the obliquity to its current value. The obliquities of the outer planets (Jupiter, Saturn, Uranus The outer planets. We also analysed the global stability of the and Neptune) are essentiallystable, and can thus be considered obliquity of the outer planets (Jupiter, Saturn, LJranus, as primordial, that is, with about the same value they had at Neptune). The orbital solution of these planets behaves in a the end of the formation of the Solar System. Nevertheless, very difterent way from those for the inner planets. As the chaotic behaviour of the obliquities under planetary perturba- difterent spectra of A( r) + tB( t) show (Fig. 2), they consist only tions could have occurred in an earlier stage of the formation of well isolated lines. There will be no large overlap of the of the Solar System, when it could have been more massive, different resonant terms as for the inner planets, and the sol- and this point should be clarified by further studies. utions are essentiallyreg.rlar. Estimates of precessionconstants The chaotic orbital motion of the inner planets8'rSresults in for the outer planets are imprecise becausetheir structure is not large-scalechaotic behaviour of their obliquity for most initial well known, but are estimated22to be well below 5" y.-t. As we conditions. The stability of the Earth's climate thus depends on obtain a global picture of the dynamics, we also learn how the the presence of the Moon which stabilizes its obliquitys, and obliquities behave for slightly difterent valuesof theseconstants. hence the insolation variations on its surface. This should be Practically speaking, it is difficult to destroy the stability of the taken into consideration when estimating the probability of obliquities until the precessionconstant of theseplanets reaches finding a planet with climate stability comparableto the Earth's 26" yr-r,where it will be in resonancewith the secularfrequency around a nearby star. n

Received26 October1992; accepted25 January1993. 18. Laskar. J. Nature 338. 237-238 (1989). (1992) 1. Harris,A. W. & Ward,W. R. A. Rev.planeL Sci 10, 61-108 (1982). 19. Davies,M. E. ef al. Celest. Mech.53,377-379 (in 2. Tremaine,S. lcarus89, 85-92 (1991). 20. Laskar, J., Joutel, F. & Boudin, F. Astr. Astrophys. the press). 3. Dones,L. & Tremaine,S. Sclence259,350-354 (1993). 27. Vlard, W. R. /carqs 50, 444-448 $982). (799I). 4. Safronov,V.S. Evolutionof the ProtoplanetaryCloud and Formation of the Earthand the Planets 22. Korycansky, D. G., Bodenheimer, P. & Pollac*, J. B. lcarus 92,234-251- (Nauka,Moscow, 1969). 23. coldreich, P. Rev.ceophys.4,411-439 (1966). (Hilger, 5. Laskar,J., Joutel, F. & Robutel,P. Nature 361, 615-617 (1993). 24. Mvtay, C. A. Vectorial Astrametry Bristol, 1983). 6. Chirikov,B.Y. Phys.Rep.52,263 (1979). 25. Laskar, ). in Chaos, Resonance and Collective Dynamical Phenomena in the Solar System (ed. (Kluwer, 7. Lambeck,K. TheEarth's VariableRotation: Geophysical Causes and Consequences(Cambridge Ferraz-Mello,S.) 1-16 Dordrecht, 1992). Univ.Press. 1980). 26. Laskar,J., Quinn, T. & Tremaine,S. /carus 95, 148-152 (1992). (1992). 8. Laskar,J. lcarus88, 266-291 (1990). 27. Sussman, G. & Wisdom, J. Science 257, 56-62 (1966). 9. MacDonald,G. ). F. Rev.Geophys. 2, 467-54I (19641. 28. Colombo,c. Astr. I 71.891-896 (1987). 10. Goldreich,P. & Soter,S. lcarus5, 375-389 (1966). 29. Henrard, J. & Murigande, C. Celest Mech. 40, 345-366 (1992). 11. Burns,J. A. lcarus28. 453-458 (1976). 30. Laskar, J. Froeschl6, C. & Celletti, A. Physica D56, 253-269 (in press). 1-2.Peale, S. J. /carus28,459-467 (1976). 31. Laskar, J. Physica D the 13. Goldreich,P. & Peale,S. J. Astr.J.75,273-284 (7970. 32. Peale,S. J. Astr. J.79,722-744 $974\. 14. Dobrovolskis,A. R. /carus41,18-35 (1980). 15. Ward,W.R. J. geophys.Res. 79, 3375-3386(1974). ACKNOWLEDGEMENTS.We thank P. Barge, A. Chenciner, S. Dumas, D. Gautier, M. Herman, 16. Hilton,). L. Astr.J. tO2, 1.570-].527$99D. F. Joutel, P. Y. Longaretti, S. Peale and B. Sicardy for discussions, and J. Marchand for technical 17. Ward,W. R.& Rudy,O. ). lcarus94, 160-164 (1991), assistance.

6L2 NATURE. VOL361 . 18 FEBRUARY1993