to give an overvie of the long-tere periodicity pary orbital otion， e perfored any fast fourier transforations ffts along the tie axis， and superposed the resultg periodgras to dra o-diensional tie–freency a the specific approach to drag these tie–freency a this paper is very siple – uch sipler than the avelet analysis or skar's 1990， 1993 freency analysis

divide the lo-pass filtered orbital data to ents of the sath the length of each data sent should be a ultiple of 2 order to apply the fft

each frant of the data has a r overppg part: for exaple， hen the ith data begs fro tti and ends at tti+t， the next data ses fro ti+δt≤ti+δt+t， here δt?t e ntue this division until e reach a certa nuber n by hich tn+t reaches the total tegration length

e apply an fft to each of the data frants， and obta n freency diagras

each freency diagra obtaed above， the strength of periodicity can be repced by a grey-scale or lour chart

e perfor the repceent， and nnect all the grey-scale or lour charts to one graph for each tegration the horizontal axis of these ne graphs should be the tie， ie the startg ties of each frant of data ti， here i 1，…， n the vertical axis represents the period or freency of the osciltion of orbital eleents

e have adopted an fft becae of its overhel speed， sce the aount of nuerical data to be deposed to freency ponents is terribly hu several tens of gbytes

a typical exaple of the tie–freency ap created by the above procedures is shon a grey-scale diagra 5， hich shos the variation of periodicity the eentricity and ction of earth n+2 tegration fig 5， the dark area shos that at the tie dicated by the vae on the abscissa， the periodicity dicated by the ordate is stronr than the lighter area around it e can regnize fro this ap that the periodicity of the eentricity and ction of earth only chans slightly over the entire period vered by the n+2 tegration this nearly regur trend is alitatively the sae other tegrations and for other ps， although typical freencies differ p by p and eleent by eleent

42 long-tere of orbital energy and angur oentu

e calcute very long-periodic variation and exchan of pary orbital energy and angur filtered deunay ele， h g and h are eivalent to the pary orbital angur oentu and its vertical ponent per unit ass l is reted to the pary orbital energy e per unit ass as e?μ22l2 if the syste is pletely lear， the orbital energy and the angur oentu each freency b t be nstant non-learity the pary syste can cae an exchan of energy and angur oentu the freency doa the aplitude of the loest-freency osciltion should crease if the syste is unstable and

eaks don gradually hoever， such a sypto of stability is not proent our long-terrations

fig 7， the total orbital energy and angur oentu of the four ner ps and all ne ps are shon for tegration n+2 the upper three panels sho the long-periodic variation of total energy denoted ase- e0， total angur o- g0， and the vertical ponent h- h0 of the ner four ps calcuted fro the lo-pass filtered deunay ele0， h0 denote the itial vaes of each antity the abte difference fro the itial vaes is plotted the panels the loer three panels each figure shoe-e0，g-g0 andh-h0 of the total of ne ps the fctuation shon the loer panels is virtually entirely a result of the assive jovian ps

the variations of energy and angur oentu of the ner four ps and all ne ps， it is apparent that the aplitudes of those of the ner ps are uch saller than those of all ne ps: the aplitudes of the outer five ps are er than those of the ner ps this does not ean that the ner terrestrial pary subsyste is ore stable than the outer one: this is siply a result of the retive sallness of the asses of the four terrestrial ps pared ith those of the outer jovian ps another thg e notice is that the ner pary subsyste ay bee unstable ore rapidly than the outer one becae of its shorter orbital tie-scales this can be seen the panels denoted asner 4 fig 7 here the lonr-periodic and irregur osciltions are ore apparent than the panels denoted astotal 9 actually， the fctuations thener 4 panels are to a r extent as a result of the orbital variation of the ercury hoever， e cannot neglect the ntribution fro other terrestrial ps， as e ill see subseent sections

44 long-ter of several neighbourg p pairs

let see e dividual variations of pary orbital energy and angur oentu expressed by the lo-pass filtered deunay eles 10 and 11 sho long-ter evotion of the orbital energy of each p and the angur oentu n+1 and n?2 tegrations e notice that e ps for apparent pairs ters of orbital energy and angur oe particur， ven and earth ake a typical pair the figures， they sho negative rretions exchan of energy and positive rretions exchan of angur oentuative rretion exchan of orbital energy eans that the o ps for a closed dynaical syste ters of the orbital energy the positive rretion exchan of angur oentu eans that the o ps are siultaneoly under certa long-ter perturbations candidates for perturbers are jupiter and saturn al fig 11， e can see that ars shos a positive rretion the angur oentu variation to the ven–earth syste ercury exhibits certa negative rretions the angur oentu vers the ven–earth syste， hich sees to be a reaction caed by the nservation of angur oentu the terrestrial pary subsyste

it is not clear at the oent hy the ven–earth pair exhibits a negative rretion energy exchan and a positive rretion angur oe e ay possibly exp this through observg the neral fact that there are no secur ters pary seiajor axes up to send-order perturbation theories cf

ouer apap cleence 1961 boaletti apap pucao 1998 this eans that the pary orbital energy hich is directly reted to the seiajor axis a ht be uch less affected by perturbg ps than is the angur oe hich retes to e hence， the eentricities of ven and earth can be disturbed easily by jupiter and saturn， hich results a positive rretion the angur oe on the other hand， the seiajor axes of ven and earth are less likely to be disturbed by the jovian ps th the energy exchan ay be liited only ith the ven–earth pair， hich results a negative rretion the exchan of orbital energy the pair

as for the outer jovian pary subsyste， jupiter–saturn and uran–neptune see to ake dynaical pairs hoever， the strength of their uplg is not as strong pared ith that of the ven–earth pair

5 ± 5 x 1010-yr tegrations of outer pary orbits

sce the jovian pary asses are er than the terrestrial pary asses， e treat the jovian pary syste as an dependent pary syste ters of the study of its dynaical stability hence， e added a uple of trial tegrations that span ± 5 x 1010 yr， cdg only the outer five ps the four jovian ps ps pto the results exhibit the rigoro stability of the outer pary syste over this long tie-span orbital nfigurations fig 12， and variation of eentricities and ctions fig 13 sho this very long-ter stability of the outer five ps both the tie and the freency doh e do not sho a here， the typical freency of the orbital osciltion of pto and the other outer ps is alost nstant durg these very long-terration periods， hich is deonstrated the tie–freency a on our ebpa

these o tegrations， the retive nuerical error the total energy as ～10?6 and that of the total angur oentu as ～10?10

51 renances the neptune–pto syste

koshita apap nakai 1996 tegrated the outer five pary orbits over ± 55 x 109 yr they found that four ajor renances beeen neptune and pto are ataed durg the hole tegration period， and that the renances ay be the a caes of the stability of the orbit of pto the ajor four renances found previo research are as follos the follog description，λ denotes the itude，Ω is the longitude of the ascendg node and ? is the longitude of perihelion subscripts p and n denote pto and neptune

ean otion renance beeen neptune and pto 3:2 the critical arguent θ1 3 λp? 2 λn??p li

ates around 180° ith an aplitude of about 80° and a li

ation period of about 2 x 104 yr

the arguent of perihelion of pto pθ2?p?Ωp li

ates around 90° ith a period of about 38 x 106 yr the doant periodic variations of the eentricity and ction of pto are synchronized ith the li

ation of its arguent of perihelion this is anticipated the secur perturbation theory nstructed by kozai 1962

the longitude of the node of pto referred to the longitude of the node of neptune，θ3Ωp?Ωn， circutes and the period of this circution is eal to the period of θ2 li

ation hen θ3 bees zero， ie the longitudes of ascendg nodes of neptune and pto overp， the ction of pto bees axiu， the eentricity bees iu and the arguent of perihelion bees 90° hen θ3 bees 180°， the ction of pto bees iu， the eentricity bees axiu and the arguent of perihelion bea illias apap benn 1971 anticipated this type of renance， ter nfired by i， nobili apap carpo 1989

an arguent θ4?p??n+ 3 Ωp?Ωn li

ates around 180° ith a long period，～ 57 x 108 yr

our nurations， the renances i–iii are ell ataed， and variation of the critical arguents θ1，θ2，θ3 rea si the hole tegration period figs 14–16 hoever， the fourth renance iv appears to be different: the critical arguent θ4 alternates li

ation and circution over a 1010-yr ti 17 this is an terestg fact that koshita apap nakai's 1995， 1996 shorter tegrations ere not able to disclose

6 discsion

hat kd of dynaical echanis atas this long-ter stability of the pary syste? e can iediately thk of o ajor features that ay be responsible for the long-ter stability first， there see to be no significant loer-order renances ean otion and secur beeen any pair a the ne ps jupiter and saturn are close to a 5:2 ean otion renance the fareat eality’， but not jt the renance zone higher-order renances ay cae the chaotic nature of the pary dynaical otion， but they are not strong as to destroy the stable pary otion ith the lifetie of the real r syste the send feature， hich e thk is ore iportant for the long-ter stability of our pary syste， is the difference dynaical distance beeen terrestrial and jovian pary subsystes ito apap tanikaa 1999， 2001 hen e easure pary separations by the utual hill radii r， separations a terrestrial ps are greater than 26rh， hereas those a jovian ps are less than 14rh this difference is directly reted to the difference beeen dynaical features of terrestrial and jovian ps terrestrial ps have saller asses， shorter orbital periods and ider dynaical separation they are strongly perturbed by jovian ps that have rr er orbital periods and narroer dynaical separation jovian ps are not perturbed by any other assive bodies

the present terrestrial pary syste is still beg disturbed by the assive jovian ps hoever， the ide separation and utual teraction a the terrestrial ps renders the disturbance effective the degree of disturbance by jovian ps is oejorder of nitude of the eentricity of jupiter， sce the disturbance caed by jovian ps is a forced osciltion havg an aplitude of oej heighteng of eentricity， for exaple oej～005， is far fro sufficient to provoke stability the terrestrial ps havg such a ide separation as 26rh th e assue that the present ide dynaical separation a terrestrial ps 26rh is probably one of the nificant nditions for the stability of the pary syste over a 109-yr tie-span our detailed analysis of the retionship beeen dynaical distance beeen ps and the stability tie-scale of r syste pary otion is no on-gog

although our nurations span the lifetie of the r syste， the nuber of tegrations is far fro sufficient to fill the itial phase space it is necessary to perfor ore and ore nurations to nfir and exae detail the long-ter stability of our pary dynaics

——以上文段引自 ito， tap tanikaa， k long-terrations and stability of pary orbits our r syste on not r astron c 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。最新网址：hx

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