g0， h0 dehe initial values of each quantity. the absolute difference from the initial values is plotted in the panels. the lower three panels in each figure showe-e0，g-g0 andh-h0 of the total of nine plas. the fluctuation shown in the lower panels is virtually entirely a result of the massive jovian plas.
paring the variations of energy and angular momentum of the inner four plas and all nine plas， it is apparent that the amplitudes of those of the inner plas are much smaller than those of all nine plas: the amplitudes of the outer five plas are much larger than those of the inner plas. this does not mean that the ierrestrial plaary subsystem is more stable thaer ohis is simply a result of the relative smallness of the masses of the four terrestrial plas pared with those of the outer jovian plas. ahiice is that the inner plaary subsystem may bee unstable more rapidly thaer one because of its shorter orbital time-scales. this be seen in the panels denoted asinner 4 in Fig. 7 where the longer-periodid irregular oscillations are more 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 ot he tribution from other terrestrial plas， as we will see in subsequeions.
4.4 Long-term coupling of several neighb pla pairs
Let us see some individual variations of plaary orbital energy and angular momentum expressed by the low-pass filtered Delaunay elements. Figs 10 and 11 show long-term evolution of the orbital energy of each pla and the angular momentum in n 1 and n?2 iions. we notice that some plas form apparent pairs in terms of orbital energy and angular momentum exge. in particular， Venus ah make a typical pair. in the figures， they show ive correlations in exge of energy and positive correlations in exge of angular momentu the ive correlation in exge of orbital energy means that the two plas form a closed dynamical system in terms of the orbital energy. the positive correlation in exge of angular momentum means that the two plas are simultaneously under certain long-term perturbations. didates for perturbers are Jupiter and saturn. also in Fig. 11， we see that mars shoositive correlation in the angular momentum variation to the Veh syste mercury exhibits certaiive correlations in the angular momentum versus the Veh system， which seems to be a rea caused by the servation of angular momentum ierrestrial plaary subsyste
it is not clear at the moment why the Veh pair exhibits a ive correlation in energy exge and a positive correlation in angular momentum exge. ossibly explain this through the general fact that there are no secular terms in plaary semimajor axes up to sed-order perturbation theories {cf. brouwer clemence 1961; boccaletti Pucacco 1998}. this means that the plaary orbital energy {which is directly related to the semimajor axis a} might be much less affected by perturbing plahan is the angular momentum exge {which relates to e}. hehe etricities of Venus ah be disturbed easily by Jupiter and saturn， which results in a positive correlation in the angular momentum exge. oher hand， the semimajor axes of Venus ah are less likely to be disturbed by the jovian plas. thus the energy exge may be limited only within the Veh pair， which results in a ive correlation in the exge of orbital energy in the pair.
as for the outer jovian plaary subsystem， Jupiter–saturn and uranus–uo make dynamical pairs. however， the strength of their coupling is not as strong pared with that of the Veh pair.
5 ± 5 × 1010-yr iions of outer plaary orbits
sihe jovian plaary masses are much larger thaerrestrial plaary masses， we treat the jovian plaary system as an indepe plaary system in terms of the study of its dynamical stability. hence， we added a couple of trial iions that span ± 5 × 1010 yr， including only the outer five plas {the four jovian plas plus Pluto}. the results exhibit the rigorous stability of the outer plaary system over this long time-span. orbital figurations {Fig. 12}， and variation of etricities and inations {Fig. 13} show this very long-term stability of the outer five plas in both the time and the frequenains. although we do not shos here， the typical frequency of the orbital oscillation of Pluto and the other outer plas is almost stant during these very long-term iion periods， which is demonstrated iime–frequency maps on our webpage.
iwo iions， the relative numerical error ial energy was ～10?6 and that of the total angular momentum was ～10?10.
5.1 Resonances in the une–Pluto system
Kinoshita nakai {1996} ied the outer five plaary orbits over ± 5.5 × 109 yr . they found that four major resonances betweeune and Pluto are maintained during the whole ii
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