对火星轨道变化问题的最后解释（2 / 2）

the variation of eentricities and orbital clations for the ner four psthe itial and fal part of the tegration n+1 is shownfig 4 as expected， the character of the variation of pary orbital elents does not differ significantly beeen the itial and fal part of each tegration， at least for ven， earth and ars the elents of rcury， especially its eentricity， see to change to a significant extent this is partly becae the orbital ti-scale of the p is the shortest of all the ps， which leads to a ore rapid orbital evotion than other ps; the nerost p ay be nearest to stability this result appears to beso agreent with skar&039;s (1994， 1996) expectations that rge and irregur variations appearthe eentricities and clations of rcury on a ti-scale of several 109 yr however， the effect of the possible stability of the orbit of rcury ay not fatally affect the global stability of the whole pary syste og to the sall ass of rcury we will ntion briefly the long-ter orbital evotion of rcury tersection 4 g low-pass filtered orbital elents

the orbital otion of the outer five ps sees rigoroly stable and quite regur over this ti-span (see also section 5)

32 ti–frequency aps

although the pary otion exhibits very long-ter stability defed as the non-existence of close enunter events， the chaotic nature of pary dynaics can change the osciltory period and aplitude of pary orbital otion gradually over such long ti-spans even such slight fctuations of orbital variationthe frequency doa， particurlythe case of earth， can potentially have a significant effect on its surface cliate syste through sor sotion variation (cf berger 1988)

to give an overview of the long-ter changeperiodicitypary orbital otion， we perford any fast fourier transforations (ffts) along the ti axis， and superposed the resultg periodgras to draw o-dinsional ti–frequency aps the specific approach to drag these ti–frequency apsthis paper is very siple – uch sipler than the wavelet analysis or skar&039;s (1990， 1993) frequency analysis

divide the low-pass filtered orbital data to any fragnts of the sa length the length of each data segnt should be a ultiple of 2order to apply the fft

each fragnt of the data has a rge overppg part: for exaple， when the ith data begs fro t=ti and ends at t=ti+t， the next data segnt ranges fro ti+δt≤ti+δt+t， where δt?t we ntue this division until we reach a certa nuber n by which tn+t reaches the total tegration length

we apply an fft to each of the data fragnts， and obta n frequency diagras

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

we perfor the repcent， and nnect all the grey-scale (or lour) charts to one graph for each tegration the horizontal axis of these new graphs should be the ti， ie the startg tis of each fragnt of data (ti， where i= 1，…， n) the vertical axis represents the period (or frequency) of the osciltion of orbital elents

we have adopted an fft becae of its overwhelg speed， sce the aount of nurical data to be deposed to frequency ponents is terribly huge (several tens of gbytes)

a typical exaple of the ti–frequency ap created by the above procedures is showna grey-scale diagra as fig 5， which shows the variation of periodicitythe eentricity and clation of earthn+2 tegrationfig 5， the dark area shows that at the ti dicated by the vae on the abscissa， the periodicity dicated by the ordate is stronger thanthe lighter area around it we can regnize fro this ap that the periodicity of the eentricity and clation of earth only changes slightly over the entire period vered by the n+2 tegration this nearly regur trend is qualitatively the saother tegrations and for other ps， although typical frequencies differ p by p and elent by elent

42 long-ter exchange of orbital energy and angur ontu

we calcute very long-periodic variation and exchange of pary orbital energy and angur ontu g filtered deunay elents l， g， h g and h are equivalent to the pary orbital angur ontu and its vertical ponent per unit ass l is reted to the pary orbital energy e per unit ass as e=?μ2/2l2 if the syste is pletely lear， the orbital energy and the angur ontueach frequency b t be nstant non-learitythe pary syste can cae an exchange of energy and angur ontuthe frequency doa the aplitude of the lowest-frequency osciltion should crease if the syste is unstable and breaks down gradually however， such a sypto of stability is not proentour long-ter tegrations

fig 7， the total orbital energy and angur ontu of the four ner ps and all ne ps are shown for tegration n+2 the upper three panels show the long-periodic variation of total energy (denoted ase- e0)， total angur ontu ( g- g0)， and the vertical ponent ( h- h0) of the ner four ps calcuted fro the low-pass filtered deunay elentse0， g0， h0 denote the itial vaes of each quantity the absote difference fro the itial vaes is plottedthe panels the lower three panelseach figure showe-e0，g-g0 andh-h0 of the total of ne ps the fctuation shownthe lower panels is virtually entirely a result of the assive jovian ps

parg the variations of energy and angur ontu 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 uch rger than those of the ner ps this does not an 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 with those of the outer jovian ps another thg we notice is that the ner pary subsyste ay bee unstable ore rapidly than the outer one becae of its shorter orbital ti-scales this can be seenthe panels denoted asner 4fig 7 where the longer-periodic and irregur osciltions are ore apparent thanthe panels denoted astotal 9 actually， the fctuationsthener 4 panels are to a rge extent as a result of the orbital variation of the rcury however， we cannot neglect the ntribution fro other terrestrial ps， as we will seesubsequent sections

44 long-ter uplg of several neighbourg p pairs

letsee so dividual variations of pary orbital energy and angur ontu expressed by the low-pass filtered deunay elents figs 10 and 11 show long-ter evotion of the orbital energy of each p and the angur ontun+1 and n?2 tegrations we notice that so ps for apparent pairsters of orbital energy and angur ontu exchangeparticur， ven and earth ake a typical pairthe figures， they show negative rretionsexchange of energy and positive rretionsexchange of angur ontu the negative rretionexchange of orbital energy ans that the o ps for a closed dynaical systeters of the orbital energy the positive rretionexchange of angur ontu ans that the o ps are siultaneoly under certa long-ter perturbations candidates for perturbers are jupiter and saturn alsofig 11， we can see that ars shows a positive rretionthe angur ontu variation to the ven–earth syste rcury exhibits certa negative rretionsthe angur ontu vers the ven–earth syste， which sees to be a reaction caed by the nservation of angur ontuthe terrestrial pary subsyste

it is not clear at the ont why the ven–earth pair exhibits a negative rretionenergy exchange and a positive rretionangur ontu exchange we ay possibly exp this through observg the general fact that there are no secur terspary seiajor axes up to send-order perturbation theories (cf brouwer ≈ap;ap;ap; clence 1961; boaletti ≈ap;ap;ap; pucao 1998) this ans that the pary orbital energy (which is directly reted to the seiajor axis a) ight be uch less affected by perturbg ps than is the angur ontu exchange (which retes to e) hence， the eentricities of ven and earth can be disturbed easily by jupiter and saturn， which resultsa positive rretionthe angur ontu exchange on the other hand， the seiajor axes of ven and earth are less likely to be disturbed by the jovian ps th the energy exchange ay be liited only with the ven–earth pair， which resultsa negative rretionthe exchange of orbital energythe pair

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

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

sce the jovian pary asses are uch rger than the terrestrial pary asses， we treat the jovian pary syste as an dependent pary systeters of the study of its dynaical stability hence， we 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 ti-span orbital nfigurations (fig 12)， and variation of eentricities and clations (fig 13) show this very long-ter stability of the outer five psboth the ti and the frequency doas although we do not show aps here， the typical frequency of the orbital osciltion of pto and the other outer ps is alost nstant durg these very long-ter tegration periods， which is deonstratedthe ti–frequency aps on our webpage

these o tegrations， the retive nurical errorthe total energy was ～10?6 and that of the total angur ontu was ～10?10

51 resonancesthe neptune–pto syste

koshita ≈ap;ap;ap; nakai (1996) tegrated the outer five pary orbits over ± 55 x 109 yrthey found that four ajor resonances beeen neptune and pto are ataed durg the whole tegration period， and that the resonances ay be the a caes of the stability of the orbit of pto the ajor four resonances foundprevio research are as followsthe follog description，λ denotes the an longitude，Ω is the longitude of the ascendg node and ? is the longitude of perihelion subscripts p and n denote pto and neptune

an otion resonance beeen neptune and pto (3:2) the critical argunt θ1= 3 λp? 2 λn??p librates around 180° with an aplitude of about 80° and a libration period of about 2 x 104 yr

the argunt of perihelion of pto wp=θ2=?p?Ωp librates around 90° with a period of about 38 x 106 yr the doant periodic variations of the eentricity and clation of pto are synchronized with the libration of its argunt of perihelion this is anticipatedthe secur perturbation theory nstructed by kozai (1962)

the longitude of the node of pto referred to the longitude of the node of neptune，θ3=Ωp?Ωn， circutes and the period of this circution is equal to the period of θ2 libration when θ3 bees zero， ie the longitudes of ascendg nodes of neptune and pto overp， the clation of pto bees axiu， the eentricity bees iu and the argunt of perihelion bees 90° when θ3 bees 180°， the clation of pto bees iu， the eentricity bees axiu and the argunt of perihelion bees 90° aga willias ≈ap;ap;ap; benson (1971) anticipated this type of resonance， ter nfird by ini， nobili ≈ap;ap;ap; carpo (1989)

an argunt θ4=?p??n+ 3 (Ωp?Ωn) librates around 180° with a long period，～ 57 x 108 yr

our nurical tegrations， the resonances (i)–(iii) are well ataed， and variation of the critical argunts θ1，θ2，θ3 rea siir durg the whole tegration period (figs 14–16 ) however， the fourth resonance (iv) appears to be different: the critical argunt θ4 alternates libration and circution over a 1010-yr ti-scale (fig 17) this is an terestg fact that koshita ≈ap;ap;ap; nakai&039;s (1995， 1996) shorter tegrations were not able to disclose

6 discsion

what kd of dynaical chanis atas this long-ter stability of the pary syste? we can idiately thk of o ajor features that ay be responsible for the long-ter stability first， there see to be no significant lower-order resonances (an otion and secur) beeen any pair aong the ne ps jupiter and saturn are close to a 5:2 an otion resonance (the fao ‘great equality’)， but not jtthe resonance zone higher-order resonances ay cae the chaotic nature of the pary dynaical otion， but they are not so strong as to destroy the stable pary otion with the lifeti of the real sor syste the send feature， which we thk is ore iportant for the long-ter stability of our pary syste， is the differencedynaical distance beeen terrestrial and jovian pary subsystes (ito ≈ap;ap;ap; tanikawa 1999， 2001) when we asure pary separations by the utual hill radii (r_)， separations aong terrestrial ps are greater than 26rh， whereas those aong 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 wider dynaical separation they are strongly perturbed by jovian ps that have rger asses， longer orbital periods and narrower 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 however， the wide separation and utual teraction aong the terrestrial ps renders the disturbance effective; the degree of disturbance by jovian ps is o(ej)(order of agnitude of the eentricity of jupiter)， sce the disturbance caed by jovian ps is a forced osciltion havg an aplitude of o(ej) heighteng of eentricity， for exaple o(ej)～005， is far fro sufficient to provoke stabilitythe terrestrial ps havg such a wide separation as 26rh th we assu that the present wide dynaical separation aong terrestrial ps (≈ap;ap;gt; 26rh) is probably one of the ost significant nditions for atag the stability of the pary syste over a 109-yr ti-span our detailed analysis of the retionship beeen dynaical distance beeen ps and the stability ti-scale of sor syste pary otion is now on-gog

although our nurical tegrations span the lifeti of the sor syste， the nuber of tegrations is far fro sufficient to fill the itial phase space it is necessary to perfor ore and ore nurical tegrations to nfir and exaedetail the long-ter stability of our pary dynaics

——以上文段引自 ito， t≈ap;ap; tanikawa， k long-ter tegrations and stability of pary orbitsour sor syste on 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。