the variation of entricities and orbital inclinations for the inner four pls in the initial and final part of the integration n1 is shown in fig. 4. as expected， the character of the variation of plary orbital elents does not differ significantly between the initial and final part of each integration， at least for venus， earth and mars. the elents of mercury， especially its entricity， seenbsp;to change to a significant extent. this is partly because the orbital tiscale of the pl is the shortest of all the pls， which leads to a re rapid orbital evolution than other pls; the innerst pl y be nearest to instability. this result appears to be in so agreent with laskar&039;s 1994， 1996 expectations that large and irregular variations appear in the entricities and inclinations of mercury on a tiscale of several 109 yr. however， the effect of the possible instability of the orbit of mercury y not fatally affect the global stability of the whole plary systenbsp;owing to the sll ss of mercury. we will ntion briefly the longternbsp;orbital evolution of mercury later in section 4 using lowpass filtered orbital elents.

the orbital tion of the outer five pls see rigorously stable and quite regular over this tispan see also section 5.

3.2 ti–frequency ps

although the plary tion exhibits very longternbsp;stability defined as the nonexistence of close encounter events， the chaotic nature of plary dynacs can change the oscillatory period and alitude of plary orbital tion gradually over such long tispans. even such slight fluctuations of orbital variation in the frequency doin， particularly in the case of earth， can potentially have a significant effect on its surface clite systenbsp;through solar insolation variation cf. berger 1988.

to give an overview of the longternbsp;change in periodicity in plary orbital tion， we perford ny fast fourier transfortions ffts along the ti axis， and superposed the resulting periodgra to draw twodinsional ti–frequency ps. the specific approach to drawing these ti–frequency ps in this paper is very sile – ch siler than the wavelet analysis or laskar&039;s 1990， 1993 frequency analysis.

divide the lowpass filtered orbital data into ny fragnts of the sa length. the length of each data segnt should be a ltiple of 2 in order to apply the fft.

each fragnt of the data has a large overlapping part: for exale， when the ith data begins fronbsp;tti and ends at ttit， the next data segnt ranges fronbsp;tiδttiδtt， where δtt. we continue this division until we reach a certain nuer n by which tnt reaches the total integration length.

we apply an fft to each of the data fragnts， and obtain n frequency diagra.

in each frequency diagranbsp;obtained above， the strength of periodicity can be replaced by a greyscale or colour chart.

we perfornbsp;the replacent， and connect all the greyscale or colour charts into one graph for each integration. the horizontal axis of these new graphs should be the ti， i.e. the starting tis of each fragnt of data ti， where i 1，…， n. the vertical axis represents the period or frequency of the oscillation of orbital elents.

we have adopted an fft because of its overwhelng speed， since the aunt of nurical data to be dposed into frequency&nbspponents is terribly huge several tens of gbytes.

a typical exale of the ti–frequency p created by the above procedures is shown in a greyscale diagranbsp;as fig. 5， which shows the variation of periodicity in the entricity and inclination of earth in n2 integration. in fig. 5， the dark area shows that at the ti indicated by the value on the abscissa， the periodicity indicated by the ordinate is stronger than in the lighter area around it. we can recognize fronbsp;this p that the periodicity of the entricity and inclination of earth only changes slightly over the entire period covered by the n2 integration. this nearly regular trend is qualitatively the sa in other integrations and for other pls， although typical frequencies differ pl by pl and elent by elent.

4.2 longternbsp;exchange of orbital energy and angular ntubr>

we calculate very longperiodic variation and exchange of plary orbital energy and angular ntunbsp;using filtered delaunay elents l， g， h. g and h are equivalent to the plary orbital angular ntunbsp;and its vertical&nbspponent per unit ss. l is related to the plary orbital energy e per unit ss as e22l2. if the systenbsp;is&nbsppletely linear， the orbital energy and the angular ntunbsp;in each frequency bin st be constant. nonlinearity in the plary systenbsp;can cause an exchange of energy and angular ntunbsp;in the frequency doin. the alitude of the lowestfrequency oscillation should increase if the systenbsp;is unstable and breaks down gradually. however， such a sytonbsp;of instability is not pronent in our longternbsp;integrations.

in fig. 7， the total orbital energy and angular ntunbsp;of the four inner pls and all nine pls are shown for integration n2. the upper three panels show the longperiodic variation of total energy denoted ase e0， total angular ntunbsp; g g0， and the vertical&nbspponenth h0 of the inner four pls calculated fronbsp;the lowpass filtered delaunay elents.e0， g0， h0 denote the initial values of each quantity. the absolute difference fronbsp;the initial values is plotted in the panels. the lower three panels in each figure showee0，gg0 andhh0 of the total of nine pls. the fluctuation shown in the lower panels is virtually entirely a result of the ssive jovian pls.

coaring the variations of energy and angular ntunbsp;of the inner four pls and all nine pls， it is apparent that the alitudes of those of the inner pls are ch sller than those of all nine pls: the alitudes of the outer five pls are ch larger than those of the inner pls. this does not an that the inner terrestrial plary subsystenbsp;is re stable than the outer one: this is sily a result of the relative sllness of the sses of the four terrestrial pls&nbsppared with those of the outer jovian pls. another thing we notice is that the inner plary subsystenbsp;y be unstable re rapidly than the outer one because of its shorter orbital tiscales. this can be seen in the panels denoted asinner 4 in fig. 7 where the longerperiodic and irregular oscillations are re apparent than in the panels denoted astotal 9. actually， the fluctuations in theinner 4 panels are to a large extent as a result of the orbital variation of the mercury. however， we cannot neglect the contribution fronbsp;other terrestrial pls， as we will see in subsequent sections.

4.4 longternbsp;coupling of several neighbouring pl pairs

let us see so individual variations of plary orbital energy and angular ntunbsp;expressed by the lowpass filtered delaunay elents. figs 10 and 11 show longternbsp;evolution of the orbital energy of each pl and the angular ntunbsp;in n1 and n2 integrations. we notice that so pls fornbsp;apparent pairs in ter of orbital energy and angular ntunbsp;exchange. in particular， venus and earth ke a typical pair. in the figures， they show negative correlations in exchange of energy and positive correlations in exchange of angular ntu the negative correlation in exchange of orbital energy ans that the two pls fornbsp;a closed dynacal systenbsp;in ter of the orbital energy. the positive correlation in exchange of angular ntunbsp;ans that the two pls are siltaneously under certain longternbsp;perturbations. candidates for perturbers are jupiter and saturn. also in fig. 11， we can see that mars shows a positive correlation in the angular ntunbsp;variation to the venus–earth syste mercury exhibits certain negative correlations in the angular ntunbsp;versus the venus–earth syste which see to be a reaction caused by the conservation of angular ntunbsp;in the terrestrial plary subsyste

it is not clear at the nt why the venus–earth pair exhibits a negative correlation in energy exchange and a positive correlation in angular ntunbsp;exchange. we y possibly explain this through observing the general fact that there are no secular ter in plary sejor axes up to secondorder perturbation theories cf. brouwer & clence 1961; baletti & puco 1998. this ans that the plary orbital energy which is directly related to the sejor axis a ght be ch less affected by perturbing pls than is the angular ntunbsp;exchange which relates to e. hence， the entricities of venus and earth can be disturbed easily by jupiter and saturn， which results in a positive correlation in the angular ntunbsp;exchange. on the other hand， the sejor axes of venus and earth are less likely to be disturbed by the jovian pls. thus the energy exchange y be lited only within the venus–earth pair， which results in a negative correlation in the exchange of orbital energy in the pair.

as for the outer jovian plary subsyste jupiter–saturn and uranus–neptune seenbsp;to ke dynacal pairs. however， the strength of their coupling is not as strong&nbsppared with that of the venus–earth pair.

551010yr integrations of outer plary orbits

since the jovian plary sses are ch larger than the terrestrial plary sses， we treat the jovian plary systenbsp;as an independent plary systenbsp;in ter of the study of its dynacal stability. hence， we added a couple of trial integrations that span51010 yr， including only the outer five pls the four jovian pls plus pluto. the results exhibit the rigorous stability of the outer plary systenbsp;over this long tispan. orbital configurations fig. 12， and variation of entricities and inclinations fig. 13 show this very longternbsp;stability of the outer five pls in both the ti and the frequency doins. although we do not show ps here， the typical frequency of the orbital oscillation of pluto and the other outer pls is alst constant during these very longternbsp;integration periods， which is denstrated in the ti–frequency ps on our webpage.

in these two integrations， the relative nurical error in the total energy was 106 and that of the total angular ntunbsp;was 1010.

5.1 resonances in the neptune–pluto systebr>

kinoshita & nakai 1996 integrated the outer five plary orbits over5.5109 yr . they found that four jor resonances between neptune and pluto are intained during the whole integration period， and that the resonances y be the in causes of the stability of the orbit of pluto. the jor four resonances found in previous research are as follows. in the following description，λ denotes the an longitude， is the longitude of the ascending node andis the longitude of perihelion. subscripts p and n denote pluto and neptune.

mean tion resonance between neptune and pluto 3:2. the critical argunt θ1 3 λp 2 λnp librates around 180 with an alitude of about 80 and a libration period of about 2104 yr.

the argunt of perihelion of pluto pθ2pp librates around 90 with a period of about 3.8106 yr. the donant periodic variations of the entricity and inclination of pluto are synchronized with the libration of its argunt of perihelion. this is anticipated in the secular perturbation theory constructed by kozai 1962.

the longitude of the node of pluto referred to the longitude of the node of neptune，θ3pn， circulates and the period of this circulation is equal to the period of θ2 libration. when θ3 bes zero， i.e. the longitudes of ascending nodes of neptune and pluto overlap， the inclination of pluto bes xi the entricity bes ninbsp;and the argunt of perihelion bes 90. when θ3 bes 180， the inclination of pluto bes ni the entricity bes xinbsp;and the argunt of perihelion bes 90 again. willia & benson 1971 anticipated this type of resonance， later confird by milani， nobili & carpino 1989.

an argunt θ4pn 3 pn librates around 180 with a long period， 5.7108 yr.

in our nurical integrations， the resonances i–iii are well intained， and variation of the critical argunts θ1，θ2，θ3 rein silar during the whole integration period figs 14–16 . however， the fourth resonance iv appears to be different: the critical argunt θ4 alternates libration and circulation over a 1010yr tiscale fig. 17. this is an interesting fact that kinoshita & nakai&039;s 1995， 1996 shorter integrations were not able to disclose.

6 discussion

what kind of dynacal chanisnbsp;intains this longternbsp;stability of the plary systenbsp;we can iediately think of two jor features that y be responsible for the longternbsp;stability. first， there seenbsp;to be no significant lowerorder resonances an tion and secular between any pair ang the nine pls. jupiter and saturn are close to a 5:2 an tion resonance the faus great inequality， but not just in the resonance zone. higherorder resonances y cause the chaotic nature of the plary dynacal tion， but they are not so strong as to destroy the stable plary tion within the lifeti of the real solar syste the second feature， which we think is re iortant for the longternbsp;stability of our plary syste is the difference in dynacal distance between terrestrial and jovian plary subsyste ito & tanikawa 1999， 2001. when we asure plary separations by the tual hill radii r， separations ang terrestrial pls are greater than 26rh， whereas those ang jovian pls are less than 14rh. this difference is directly related to the difference between dynacal features of terrestrial and jovian pls. terrestrial pls have sller sses， shorter orbital periods and wider dynacal separation. they are strongly perturbed by jovian pls that have larger sses， longer orbital periods and narrower dynacal separation. jovian pls are not perturbed by any other ssive bodies.

the present terrestrial plary systenbsp;is still being disturbed by the ssive jovian pls. however， the wide separation and tual interaction ang the terrestrial pls renders the disturbance ineffective; the degree of disturbance by jovian pls is oejorder of gnitude of the entricity of jupiter， since the disturbance caused by jovian pls is a forced oscillation having an alitude of oej. heightening of entricity， for exale oej0.05， is far fronbsp;sufficient to provoke instability in the terrestrial pls having such a wide separation as 26rh. thus we assu that the present wide dynacal separation ang terrestrial pls > 26rh is probably one of the st significant conditions for intaining the stability of the plary systenbsp;over a 109yr tispan. our detailed analysis of the relationship between dynacal distance between pls and the instability tiscale of solar systenbsp;plary tion is now ongoing.

although our nurical integrations span the lifeti of the solar syste the nuer of integrations is far fronbsp;sufficient to fill the initial phase space. it is necessary to perfornbsp;re and re nurical integrations to confirnbsp;and exane in detail the longternbsp;stability of our plary dynacs.

以上文段引自 ito， t.& tanikawa， k. longternbsp;integrations and stability of plary orbits in our solar syste mon. not. r. astron. soc. 336， 483–500 2002

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