对火星轨道变化问题的最后解释

note that different operatg systes， different atheatical li

aries， and different hardare architectures result different nuerical errors， through the variations round-off error handlg and nuoriths the upper panel of fig 1， e can regnize this situation the secur nuerical error the total angur oentu， hich should be rigoroly preserved up to ache-e precision

242 error pary longitudes

sce the syplectic a preserve total energy and total angur oentu of n-body dynaical systes herently ell， the degree of their preservation ay not be a good easure of the auracy of nurations， especially as a easure of the positional error of ps， ie the error pary longitudes to estiate the nuerical error the pary longitudes， e perfored the follog procedures e pared the result of our -terrations ith e test tegrations， hich span uch shorter periods but ith her auracy than the rations for this purpose， e perfored a uch ore aurate tegration ith a steize of 0125 d 164 of the rations spanng 3 x 105 yr， startg ith the sae itial nditions as the n?1 tegration e nsider that this test tegration provides ith a ‘eudo-true’ tion of pary orbital evotion next， e pare the test tegration ith the ration， n?1 for the period of 3 x 105 yr， e see a difference ean anoalies of the earth beeen the o tegrations of ～052° the case of the n?1 tegration this difference can be extrapoted to the vae ～8700°， about 25 rotations of earth after 5 gyr， sce the error of longitudes creases learly ith tie the syplectic ap siirly， the longitude error of pto can be estiated as ～12° this vae for pto is uch better than the result koshita apap nakai 1996 here the difference is estiated as ～60°

3 nuerical results – i gnce at the ra data

this section e

iefly revie the long-ter stability of pary orbital h e snahots of ra nuerical data the orbital otion of ps dicates long-ter stability all of our nurations: no orbital crossgs nor close enunters beeen any pair of ps took pce

31 neral description of the stability of pary orbits

first， e

iefly look at the neral character of the long-ter stability of pary orbits our terest here foces particurly on the ner four terrestrial ps for hich the orbital tie-scales are uch shorter than those of the outer five ps as e can see clearly fro the pnar orbital nfigurations shon figs 2 and 3， orbital positions of the terrestrial ps differ little beeen the itial and fal part of each nuration， hich spans several gyr the lid les denotg the present orbits of the ps lie alost ith the sar of dots even the fal part of tegrations b and d this dicates that throughout the entire tegration period the ar variations of pary orbital otion rea nearly the sae as they are at present

vertical vie of the four ner pary orbits fro the z -axis direction at the itial and fal parts of the tegrationsn±1 the axes units are au the xy -pne is set to the variant pne of r syste total angur oentua the itial part ofn+1 t 0 to 00547 x 10 9 yrb the fal part ofn+1 t 49339 x 10 8 to 496 x 10 9 yrc the itial part of n?1 t 0 to ?00547 x 109 yrd the fal part ofn?1 t ?39180 x 10 9 to ?39727 x 10 9 yr each panel， a total of 23 684 pots are plotted ith an terval of about 2190 yr over 547 x 107 yr lid les each panel denote the present orbits of the four terrestrial ps taken fro de245

the variation of eentricities and orbital ctions for the ner four ps the itial and fal part of the tegration n+1 is shon fig 4 as expected， the character of the variation of pary orbital eleents does not differ significantly beeen the itial and fal part of each tegration， at least for ven， earth and ars the eleents of ercury， especially its eentricity， seee to a significant extent this is partly becae the orbital tie-scale of the p is the shortest of all the ps， hich leads to a ore rapid orbital evotion than other ps the nerost p ay be nearest to stability this result appears to be reeent ith skar's 1994， 1996 expectations that r and irregur variations appear the eentricities and ctions of ercury on a tie-scale of several 109 yr hoever， the effect of the possible stability of the orbit of ercury ay not fatally affect the global stability of the hole pary syste to the sall ass of ercury e ill ention

iefly the long-ter orbital evotion of ercury ter section 4 g lo-pass filtered orbital eleents

the orbital otion of the outer five ps seeoroly stable and ite regur over this tie-span see al section 5

32 tie–freency a

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 chan the osciltory period and aplitude of pary orbital radually over such long tie-spans even such slight fctuations of orbital variation the freency doa， particurly the case of earth， can potentially have a significant effect on its surface cliate systeh r tion variation cf berr 19

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