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The Chaotic Obliquity of the Planets

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The Chaotic Obliquity of the Planets ARTICLES The chaoticobliquity of the planets J. Laskar& P. Robutel Astronomieet SystdmesDynamiques, Bureau des Longitudes,TT Avenue Denfert-Rochereau, F-75014 Paris, 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 Mars 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 (Mercury, Venus, the Earth 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 precessionin 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.
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