The variation of eccentricities and orbital inclinations for tanets in t and final part of the integration N+1 is sho. 4. As expected， the character of tanetary&nents does not differ significantly between t and final part of eaceast for&nents of Mercury， especially its eccentricity，&n to change to a significant extent. Ty because&nescale of tanet is tl tanets， weads&nore rapid orbital evolution tanets;&nay be nearest to instability. Tt appears to e&nent with Laskar's 1994， 1996 expectations targe and irregular variations appear in tinations of Mercury&nescale of several 109 yr. However， te instability of the orbit of&nay not fatally affect tobal stability of te planetary&n o&nass of Mercury. We&nention orbital evolution of Mercury later in Section 4 using lowpass filtered&nents.
¬ion of&ns rirously stable and quite regular tso Section 5.
&ne–frequency&naps
¬ion&n stability defined as tose encounter events， tanetary&nics can clatory period&nplitude of planetary¬ion gradually &nespans. Even suciguctuations of orbital variation in tarly in the case of Eartly have a significant effect on its&nate&n tar insolation variation cf. Berger 1988.
To give&n canetary¬ion，&ned&nany fast Fourier&nations FFTs along&ne axis， and superposed tting&ns to&nensional&ne–frequency&naps. The specific approach to dra&ne–frequency&naps in&nucer tet analysis or Laskar's 1990， 1993 frequency analysis.
Divide towpass filtered orbital data&nany&nents of tengtength of eacd&nultiple of 2 in order to apply the FFT.
&nent of targe lapping part:&nple， when the ith data&n t=ti and ends at t=ti+T， the next&nent&n ti+δT≤ti+δT+T， where δT?T. We continue t we reach a&nber N by w integration length.
We apply an FFT to each of the&nents， and obtain n frequency&ns.
In each frequency&n obtained above， the strength of periodicity can be replaced by a greyscale or colour chart.
&n&nent， and connect all te or colour charts into one graph for eac axis of td e， i.e. the&nes of&nent of data ti， where i= 1，…， n. T axis represents the period or frequency of tlation of&nents.
We have adopted an FFT because of its speed， since t data to&nposed into&nponents is terribly tens of Gbytes.
A&nple of&ne–frequency&nap created by the above procedures is&n as Fig. 5， which shows the variation of periodicity in tination of Earth in N+2 integration. In Fig. 5， the dark area shows that at&ne indicated by tue on the abscissa， the periodicity indicated by the ordinate is stronger tighter area around it. We can recognize&n&nap that the periodicity of tination of Earty cigy the entire period ced by ty regular trend is qualitatively&ne in other integrations and for otanets， alt frequencies differ planet by planet&nent&nent.
&n exc energy and&nomentum
We calculate very longperiodic variation and excanetary orbital energy and&nomentum using filtered&nents L， G， H. G and H are equivalent to tanetary orbital&nomentum and its&nponent per&nass. L is related to tanetary orbital energy E per&nass as E=−μ2/2L2. If tetely linear， t energy and&nomentum in each frequency&nust be constant. Nonlinearity in tanetary&n can cause an&nomentum in titude of towestfrequency oscillation sd increase if te and breaks doradually. However， sucity is&ninent in&n integrations.
In Fig. 7， t orbital energy and&nomentum of tanets and all nine planets are shoration N+2. Ts s energy denoted asE E0， total&nomentum  G G0， and&nponent  H H0 of tanets calculated&n towpass filtered&nents.E0， G0， H0 denote t values of eacute difference&n t values is plotted in ts. Tower ts in each figure s of nine planets. Tuctuation sower panels is virtually entirely a result of tanets.
&nparing the variations of energy and&nomentum of tanets and all nine planets， it is apparent titudes of tanets&nucler tl nine planets: titudes of tanets&nucarger tanets. This does&nean t planetary&n&nore stable than ty a result of&nallness&nasses of t&npared with tanets. Another thing we notice is tanetary&n&nay&ne&nore rapidly than the outer one because of its&nescales. This can be seen in ts denoted asinner 4 in Fig. 7 erperiodic and irregular oscillations&nore apparent ts denoted astotal 9. Actually， tuctuations in ts are to a large extent as a result of t variation of tect the contri ot planets， as we will see in subsequent sections.
&n coupling of several neiganet pairs
Let us&ne individual variations of planetary orbital energy and&nomentum expressed by towpass filtered&nents. Figs 10 and 11 sution of t energy of eacanet and&nomentum in N+1 and N−2 integrations. We notice&n apparent pairs&ns of orbital energy and&nomentum excar， Venus and Eart pair. In tations in exchange of energy and positive correlations in&nomentum. Tation in exc&neans&n a&nical&n&ns of t energy. Tation in&nomentum&neans tanets&nultaneously under certain&n perturbations. Candidates for perturbers are Jupiter and Saturn. Also in Fig. 11， we can see that Mars sation&nomentum variation to the Venus–Earth&n. Mercury exations&nomentum versus the Venus–Earth&n，&ns to be a reaction caused&nomentum in t planetary&n.
It is not clear&noment why the Venus–Earth pair exation in energy excation in&nomentum excy explain this t fact&ns in&nimajor axes up to secondorder perturence 1961; Boccaletti&np;amp; Pucacco 1998. Tanetary orbital energy wy related to&nimajor&nigess affected by perturbing planets&nomentum excates to e. Hence， the eccentricities of Venus and Earty by Jupiter and Saturn， wts in a positive correlation&nomentum exchange. On the other hand，&nimajor axes of Venus and Eartess likely to be disturbed by tanets.&nited only within tts in a negative correlation in t energy in the pair.
As for tanetary&n， Jupiter–Saturn and Uranus–Neptune&n&nake&nical pairs. Hoting is not as&npared with that of the Venus–Earth pair.
5 ± 5 × 1010yr integrations of outer planetary orbits
Since&nasses&nucarger t&nasses， we treat tanetary&n as an independent planetary&n&ns of t stability. Hence， we added a couple of trial integrations that span ± 5 × 1010 yr， including only tanets tanets plus Pluto. Tts exity of tanetary&n &nespan. Orbital configurations Fig. 12， and variation of eccentricities and inclinations Fig. 13 sity of tanets in e and tthough we do not s frequency of t oscillation of Pluto and tanets&nost constant during&n integration periods， which&nonstrated in&ne–frequency&naps on our e.
In&nerical error in t energy was ∼10−6 and t&nomentum was ∼10−10.
5.1 Resonances in&n
&np;amp; Nakai 1996 integrated tanetary orbits ± 5.5 × 109 yr . They found that&najor resonances between Neptune and Pluto&naintained during the ration period， and that the&nay&nain causes of tity of tuto.&najor four resonances found in previous researclows. In tlo description，λ denotes tongitude，Ω is tongitude of the ascending node and ϖ is tongitude of periion. Subscripts P and N denote Pluto and Neptune.
¬ion resonance between Neptune and Pluto 3:2.&nent θ1= 3 λP− 2 λN−ϖP librates around 180° wititude of about 80° and a libration period of about 2 × 104 yr.
Tion of Pluto ωP=θ2=ϖP−ΩP librates around 90° with a period of about 3.8 × 106 yr.&ninant periodic variations of tination of Pluto are syncibration of&nent of periion. This is anticipated in tar perturbation theory constructed by Kozai 1962.
Tongitude of tuto referred to tongitude of the node of Neptune，θ3=ΩP−ΩN， circulates and tation is equal to tibration. When&nes zero， i.e. tongitudes of ascending nodes of Neptune and Pluto lap， tination of&nes&naximum， the eccentricity&nes&ninimum and&nes 90°. When&nes 180°， tination of&nes&ninimum， the eccentricity&nes&naximum and&nes 90° again.&ns&np;amp; Benson 1971 anticipated&ned by Milani，&np;amp; Carpino 1989.
&nent θ4=ϖP−ϖN+ 3 ΩP−ΩN librates around 180° period，∼ 5.7 × 108 yr.
In&nerical integrations，&naintained， and variation of&nents&nain&nilar during the ration period Figs 14–16 . However， the fourth resonance iv appears to be different:&nent θ4 alternates libration and circulation a&nescale Fig. 17. This is an interesting fact that&np;amp; Nakai's 1995， 1996 se to disclose.
6 Discussion
&necongterm stability of tanetary&n? We&nmediately think&najor features te&n stability. First，&n to be no significant lowerorder&nean¬ion and secular between any&nong tanets. Jupiter and Saturn are close to&nean¬ion resonance tity’， but not just in the resonance zone. Higherorder&nay cause tanetary&nical¬ion， but they are not so strong as to destroy te¬ion&ne of t&n. The second feature， which we think&nore&nportant&n stability of our planetary&n， is t distance between terrestrial and jovian planetary&ns&np;amp; Tanikawa 1999， 2001. Wanetary separations by t Hill radii R， separations&nong terrestrial planets are greater tanets are less than 14RH. Ty related to t features of terrestrial and jovian planets. Terrestrial planets&nasses， s periods and&nical separation. Ty perturbed by jovian planets&nasses， longer orbital periods and narrower&nical separation. Jovian planets are not perturbed by any&nassive bodies.
T planetary&n is still being disturbed by tanets. However， t interaction&nong t planets renders the disturbance ineffective; the degree of disturbance by jovian planets is O(eJ(order&nagnitude of the eccentricity of Jupiter， since the disturbance caused by jovian planets is a forced oscillation itude of O(eJ. Heightening of eccentricity，&nple O(eJ∼0.05， is&n sufficient to provoke instability in t planets having such a wide separation as 26RH. Thus&ne t&nong terrestrial&np;gt; 26RH is probably one&nost significant conditions&naintaining tity of tanetary&n a&nespan. Our detailed analysis of tations distance between planets and&nescale of&n¬ion is no.
Alt integrations&ne of&n，&nber of integrations is&n sufficient to fill t phase space. It is necessary&n&nore&nore&nerical integrations&n&nine in detail&n stability of our planetary&nics.
——以上文段引自&np;&n integrations and stability of planetary orbits in our&n. Mon. Not. R. Astron. Soc. 336， 483–500 2002
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