第一百日（1）作屎的老猫（第9/10页）

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.

The present terrestrial planetary system is still being disturbed by the massive jovian planets. However， the wide separation and mutual interaction among the terrestrial planets renders the disturbance ineffective; the degree of disturbance by jovian planets is O(eJ)(order of magnitude of the eccentricity of Jupiter)， since the disturbance caused by jovian planets is a forced oscillation having an amplitude of O(eJ). Heightening of eccentricity， for example O(eJ)～0.05， is far from sufficient to provoke instability in the terrestrial planets having such a wide separation as 26RH. Thus we assume that the present wide dynamical separation among terrestrial planets (> 26RH) is probably one of the most significant conditions for maintaining the stability of the planetary system over a 109-yr time-span. Our detailed analysis of the relationship between dynamical distance between planets and the instability time-scale of Solar system planetary motion is now on-going.