第一百日（1）作屎的老猫（第7/8页）

As for the outer jovian planetary subsystem， Jupiter–Saturn and Uranus–Neptune seem to make dynamical pairs. However， the strength of their coupling is not as strong compared with that of the Venus–Earth pair.

5 ± 5 × 1010-yr integrations of outer planetary orbits

Since the jovian planetary masses are much larger than the terrestrial planetary masses， we treat the jovian planetary system as an independent planetary system in terms of the study of its dynamical stability. Hence， we added a couple of trial integrations that span ± 5 × 1010 yr， including only the outer five planets (the four jovian planets plus Pluto). The results exhibit the rigorous stability of the outer planetary system over this long time-span. Orbital configurations (Fig. 12)， and variation of eccentricities and inclinations (Fig. 13) show this very long-term stability of the outer five planets in both the time and the frequency domains. Although we do not show maps here， the typical frequency of the orbital oscillation of Pluto and the other outer planets is almost constant during these very long-term integration periods， which is demonstrated in the time–frequency maps on our webpage.

In these two integrations， the relative numerical error in the total energy was ～10?6 and that of the total angular momentum was ～10?10.

5.1 Resonances in the Neptune–Pluto system

Kinoshita & Nakai (1996) integrated the outer five planetary orbits over ± 5.5 × 109 yr . They found that four major resonances between Neptune and Pluto are maintained during the whole integration period， and that the resonances may be the main causes of the stability of the orbit of Pluto. The major four resonances found in previous research are as follows. In the following description，λ denotes the mean longitude，Ω is the longitude of the ascending node and ? is the longitude of perihelion. Subscripts P and N denote Pluto and Neptune.

Mean motion resonance between Neptune and Pluto (3:2). The critical argument θ1= 3 λP? 2 λN??P librates around 180° with an amplitude of about 80° and a libration period of about 2 × 104 yr.

The argument of perihelion of Pluto ωP=θ2=?P?ΩP librates around 90° with a period of about 3.8 × 106 yr. The dominant periodic variations of the eccentricity and inclination of Pluto are synchronized with the libration of its argument 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，θ3=ΩP?ΩN， circulates and the period of this circulation is equal to the period of θ2 libration. When θ3 becomes zero， i.e. the longitudes of ascending nodes of Neptune and Pluto overlap， the inclination of Pluto becomes maximum， the eccentricity becomes minimum and the argument of perihelion becomes 90°. When θ3 becomes 180°， the inclination of Pluto becomes minimum， the eccentricity becomes maximum and the argument of perihelion becomes 90° again. Williams & Benson (1971) anticipated this type of resonance， later confirmed by Milani， Nobili & Carpino (1989).

An argument θ4=?P??N+ 3 (ΩP?ΩN) librates around 180° with a long period，～ 5.7 × 108 yr.

In our numerical integrations， the resonances (i)–(iii) are well maintained， and variation of the critical arguments θ1，θ2，θ3 remain similar during the whole integration period (Figs 14–16 ). However， the fourth resonance (iv) appears to be different: the critical argument θ4 alternates libration and circulation over a 1010-yr time-scale (Fig. 17). This is an interesting fact that Kinoshita & Nakai's (1995， 1996) shorter integrations were not able to disclose.

6 Discussion

What kind of dynamical mechanism maintains this long-term stability of the planetary system? We can immediately think of two major features that may be responsible for the long-term stability. First， there seem to be no significant lower-order resonances (mean motion and secular) between any pair among the nine planets. Jupiter and Saturn are close to a 5:2 mean motion resonance (the famous ‘great inequality’)， but not just in the resonance zone. Higher-order resonances may cause the chaotic nature of the planetary dynamical motion， but they are not so strong as to destroy the stable planetary motion within the lifetime of the real Solar system. The second feature， which we think is more important for the long-term stability of our planetary system， is the difference in dynamical distance between terrestrial and jovian planetary subsystems (Ito & Tanikawa 1999， 2001). When we measure planetary separations by the mutual Hill radii (R_)， separations among terrestrial planets are greater than 26RH， whereas those among jovian planets are less than 14RH. This difference is directly related to the difference between dynamical features of terrestrial and jovian planets. Terrestrial planets have smaller masses， shorter orbital periods and wider dynamical separation. They are strongly perturbed by jovian planets that have larger masses， longer orbital periods and narrower dynamical separation. Jovian planets are not perturbed by any other massive bodies.