on 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 dehe mean longitude，Ω is the longitude of the asding node and ? is the longitude of perihelion. subscripts P and e Pluto aune.
mean motion resoweeune and Pluto {3:2}. the critical argument θ1 3 λP? 2 λnP librates around 180° with an amplitude of about 80° and a libration period of about 2 × 104 yr.
the argument of perihelion of Pluto ωPθ2?P?ΩP librates around 90° with a period of about 3.8 × 106 yr. the dominant periodic variations of the etricity and ination of Pluto are synized with the libration of its argument of perihelion. this is anticipated in the secular perturbation theory structed by Kozai {1962}.
the longitude of the node of Pluto referred to the longitude of the node of une，θ3ΩP?Ωn， circulates and the period of this circulation is equal to the period of θ2 libration. when θ3 bees zero， i.e. the longitudes of asding nodes of une and Pluto overlap， the ination of Pluto bees maximum， the etricity bees minimum and the argument of perihelion bees 90°. when θ3 bees 180°， the ination of Pluto bees minimum， the etricity bees maximum and the argument of perihelion bees 90° again. williams benson {1971} anticipated this type of resonance， later firmed by milani， nobili carpino {1989}.
an argument θ4?Pn 3 {ΩP?Ωn} librates around 180° with a long period，～ 5.7 × 108 yr.
in our numerical iions， the resonances {i}–{iii} are well maintained， and variation of the critical arguments θ1，θ2，θ3 remain similar during the whole iion 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 iing fact that Kinoshita nakai“s {1995， 1996} shorter iions were not able to disclose.
6 Discussion
what kind of dynamical meism maintains this long-term stability of the plaary system? we immediately think of two major features that may be responsible for the long-term stability. First， there seem to be no signifit lower-order resonances {mean motion and secular} between any pair among the nine plas. Jupiter and saturn are close to a 5:2 mean motion resohe famous ‘great inequality’}， but not just in the resonance zone. higher-order resonances may cause the chaotiature of the plaary dynamical motion， but they are not s as to destroy the stable plaary motion within the lifetime of the real solar syste the sed feature， which we think is more important for the long-term stability of our plaary system， is the differen dynamical distaween terrestrial and jovian plaary subsystems {ito tanikawa 1999， 2001}. when we measure plaary separations by the mutual hill radii {R_}， separations among terrestrial plas are greater than 26Rh， whereas those among jovian plas are less than 14Rh. this difference is directly related to the differeween dynamical features of terrestrial and jovian plas. terrestrial plas have smaller masses， shorter orbital periods and wider dynamical separation. they are strongly perturbed by jovian plahat have larger masses， longer orbital periods and narrower dynamical separation. Jovian plas are not perturbed by any other massive bodies.
the present terrestrial plaary system is still being disturbed by the massive jovian plas. however， the wide separation and mutual iion among the terrestrial plas rehe disturbaneffective; the degree of disturbance by jovian plas is o{eJ}{order of magnitude of the etricity of Jupiter}， sihe disturbance caused by jovian plas is a forced oscillation having an amplitude of o{eJ}. heightening of etricity， for example o{eJ}～0.05， is far from suffit to provoke instability ierrestrial plas having such a wide separation as 26Rh. thus we assume that the present wide dynamical separation among terrestrial plas {gt; 26Rh} is probably one of the most signifit ditions for maintaining the stability of the plaary system over a 109-yr time-span. our detailed analysis of the relationship between dynamical distaween plas and the instability time-scale of solar system plaary motion is now on-going.
although our numerical iions span the lifetime of the solar system， the number of iions is far from suffit to fill the initial phase space. it is necessary to perform more and more numerical iions to firm and examine iail the long-term stability of our plaary dynamics.
——以上文段引自 ito， t. tanikawa， K. Long-term iions and stability of plaary orbits in our solar syste mon. not. R. astron. soc. 336， 483–500 {2002}
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