es 90° again. williams & benson (1971) anticipated this type of resonance， later confirmed&nbilani， nobili & carpino (1989).
an argument θ4=?p??n+ 3 (Ωp?Ωn) librates around 180° with a long period，～ 5.7 x 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&nb maintains this&nb stability of the planetary system? we can immediately think of two major features that may be responsible for the&nb 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&nbore important for the&nb stability of our planetary system， is the difference in dynamical distance between terrestrial and jovian planetary&nbs (ito & tanikawa 1999， 2001). when&nbeasure 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&nbagnitude 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 e 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.
although our numerical integrations span the lifetime of the solar system， the number of integrations is far from sufficient to fill the initial phase space. it is necessary to perform more and more numerical integrations to confirm and examine in detail the&nb stability of our planetary dynamics.
——以上文段引自 ito，amp; tanikawa， k.&nb integrations and stability of planetary orbits in our solar syon. not. r. astron. soc. 336， 483–500 (2002)
这只是作者君参考的一篇文章，关于太阳系的稳定性。
还有其他论文，不过也都是英文的，相关课题的中文文献很少，那些论文下载一篇要九美元（《nature》真是暴利），作者君写这篇文章的时候已经回家，不在检测中心，所以没有数据库的使用权，下不起，就不贴上来了。