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以下是文章内容：

Long-tr ntgratons and stalty of plantary orts n our Solar syst

Astract

W prsnt th rsults of vry long-tr nurcal ntgratons of plantary ortal otons ovr 109 -yr t-spans ncludng all nn plants. A quck nspcton of our nurcal data shos that th plantary oton， at last n our spl dynacal odl， ss to qut stal vn ovr ths vry long t-span. A closr look at th lost-frquncy oscllatons usng a lo-pass fltr shos us th potntally dffusv charactr of trrstral plantary oton， spcally that of Mrcury. Th havour of th ccntrcty of Mrcury n our ntgratons s qualtatvly slar to th rsults fro Jacqus Laskar's scular prturaton thory (.g. ax～ 0.35 ovr ～± 4 Gyr). Hovr， thr ar no apparnt scular ncrass of ccntrcty or nclnaton n any ortal lnts of th plants， hch ay rvald y stll longr-tr nurcal ntgratons. W hav also prford a coupl of tral ntgratons ncludng otons of th outr fv plants ovr th duraton of ± 5 × 1010 yr. Th rsult ndcats that th thr ajor rsonancs n th Nptun–Pluto syst hav n antand ovr th 1011-yr t-span.

1 Introducton

1.1Dfnton of th prol

Th quston of th stalty of our Solar syst has n datd ovr svral hundrd yars， snc th ra of Nton. Th prol has attractd any faous athatcans ovr th yars and has playd a cntral rol n th dvlopnt of non-lnar dynacs and chaos thory. Hovr， do not yt hav a dfnt ansr to th quston of hthr our Solar syst s stal or not. Ths s partly a rsult of th fact that th dfnton of th tr ‘stalty’ s vagu hn t s usd n rlaton to th prol of plantary oton n th Solar syst. Actually t s not asy to gv a clar， rgorous and physcally anngful dfnton of th stalty of our Solar syst.

Aong any dfntons of stalty， hr adopt th Hll dfnton (Gladan 1993): actually ths s not a dfnton of stalty， ut of nstalty. W dfn a syst as cong unstal hn a clos ncountr occurs sohr n th syst， startng fro a crtan ntal confguraton (Chars， Wthrll ≈ap;ap;ap; Boss 1996; Ito ≈ap;ap;ap; Tankaa 1999). A syst s dfnd as xprncng a clos ncountr hn to ods approach on anothr thn an ara of th largr Hll radus. Othrs th syst s dfnd as ng stal. Hncforard stat that our plantary syst s dynacally stal f no clos ncountr happns durng th ag of our Solar syst， aout ±5 Gyr. Incdntally， ths dfnton ay rplacd y on n hch an occurrnc of any ortal crossng tn thr of a par of plants taks plac. Ths s caus kno fro xprnc that an ortal crossng s vry lkly to lad to a clos ncountr n plantary and protoplantary systs (Yoshnaga， Kokuo ≈ap;ap;ap; Makno 1999). Of cours ths statnt cannot sply appld to systs th stal ortal rsonancs such as th Nptun–Pluto syst.

1.2Prvous studs and as of ths rsarch

In addton to th vagunss of th concpt of stalty， th plants n our Solar syst sho a charactr typcal of dynacal chaos (Sussan ≈ap;ap;ap; Wsdo 1988， 1992). Th caus of ths chaotc havour s no partly undrstood as ng a rsult of rsonanc ovrlappng (Murray ≈ap;ap;ap; Holan 1999; Lcar， Frankln ≈ap;ap;ap; Holan 2001). Hovr， t ould rqur ntgratng ovr an nsl of plantary systs ncludng all nn plants for a prod covrng svral 10 Gyr to thoroughly undrstand th long-tr voluton of plantary orts， snc chaotc dynacal systs ar charactrzd y thr strong dpndnc on ntal condtons.

Fro that pont of v， any of th prvous long-tr nurcal ntgratons ncludd only th outr fv plants (Sussan ≈ap;ap;ap; Wsdo 1988; Knoshta ≈ap;ap;ap; Naka 1996). Ths s caus th ortal prods of th outr plants ar so uch longr than thos of th nnr four plants that t s uch asr to follo th syst for a gvn ntgraton prod. At prsnt， th longst nurcal ntgratons pulshd n journals ar thos of Duncan ≈ap;ap;ap; Lssaur (1998). Although thr an targt as th ffct of post-an-squnc solar ass loss on th stalty of plantary orts， thy prford any ntgratons covrng up to ～1011 yr of th ortal otons of th four jovan plants. Th ntal ortal lnts and asss of plants ar th sa as thos of our Solar syst n Duncan ≈ap;ap;ap; Lssaur's papr， ut thy dcras th ass of th Sun gradually n thr nurcal xprnts. Ths s caus thy consdr th ffct of post-an-squnc solar ass loss n th papr. Consquntly， thy found that th crossng t-scal of plantary orts， hch can a typcal ndcator of th nstalty t-scal， s qut snstv to th rat of ass dcras of th Sun. Whn th ass of th Sun s clos to ts prsnt valu， th jovan plants ran stal ovr 1010 yr， or prhaps longr. Duncan ≈ap;ap;ap; Lssaur also prford four slar xprnts on th ortal oton of svn plants (Vnus to Nptun)， hch covr a span of ～109 yr. Thr xprnts on th svn plants ar not yt coprhnsv， ut t ss that th trrstral plants also ran stal durng th ntgraton prod， antanng alost rgular oscllatons.

On th othr hand， n hs accurat s-analytcal scular prturaton thory (Laskar 1988)， Laskar fnds that larg and rrgular varatons can appar n th ccntrcts and nclnatons of th trrstral plants， spcally of Mrcury and Mars on a t-scal of svral 109 yr (Laskar 1996). Th rsults of Laskar's scular prturaton thory should confrd and nvstgatd y fully nurcal ntgratons.

In ths papr prsnt prlnary rsults of sx long-tr nurcal ntgratons on all nn plantary orts， covrng a span of svral 109 yr， and of to othr ntgratons covrng a span of ± 5 × 1010 yr. Th total lapsd t for all ntgratons s or than 5 yr， usng svral ddcatd PCs and orkstatons. On of th fundantal conclusons of our long-tr ntgratons s that Solar syst plantary oton ss to stal n trs of th Hll stalty ntond aov， at last ovr a t-span of ± 4 Gyr. Actually， n our nurcal ntgratons th syst as far or stal than hat s dfnd y th Hll stalty crtron: not only dd no clos ncountr happn durng th ntgraton prod， ut also all th plantary ortal lnts hav n confnd n a narro rgon oth n t and frquncy doan， though plantary otons ar stochastc. Snc th purpos of ths papr s to xht and ovrv th rsults of our long-tr nurcal ntgratons， sho typcal xapl fgurs as vdnc of th vry long-tr stalty of Solar syst plantary oton. For radrs ho hav or spcfc and dpr ntrsts n our nurcal rsults， hav prpard a pag (accss )， hr sho ra ortal lnts， thr lo-pass fltrd rsults， varaton of Dlaunay lnts and angular ontu dfct， and rsults of our spl t–frquncy analyss on all of our ntgratons.

In Scton 2 rfly xplan our dynacal odl， nurcal thod and ntal condtons usd n our ntgratons. Scton 3 s dvotd to a dscrpton of th quck rsults of th nurcal ntgratons. Vry long-tr stalty of Solar syst plantary oton s apparnt oth n plantary postons and ortal lnts. A rough staton of nurcal rrors s also gvn. Scton 4 gos on to a dscusson of th longst-tr varaton of plantary orts usng a lo-pass fltr and ncluds a dscusson of angular ontu dfct. In Scton 5， prsnt a st of nurcal ntgratons for th outr fv plants that spans ± 5 × 1010 yr. In Scton 6 also dscuss th long-tr stalty of th plantary oton and ts possl caus.

2 Dscrpton of th nurcal ntgratons

（本部分涉及比较复杂的积分计算，作者君就不贴上来了，贴上来了起点也不一定能成功显示。）

2.3 Nurcal thod

W utlz a scond-ordr Wsdo–Holan syplctc ap as our an ntgraton thod (Wsdo ≈ap;ap;ap; Holan 1991; Knoshta， Yoshda ≈ap;ap;ap; Naka 1991) th a spcal start-up procdur to rduc th truncaton rror of angl varals，‘ar start’(Saha ≈ap;ap;ap; Tran 1992， 1994).

Th stpsz for th nurcal ntgratons s 8 d throughout all ntgratons of th nn plants (N±1，2，3)， hch s aout 1/11 of th ortal prod of th nnrost plant (Mrcury). As for th dtrnaton of stpsz， partly follo th prvous nurcal ntgraton of all nn plants n Sussan ≈ap;ap;ap; Wsdo (1988， 7.2 d) and Saha ≈ap;ap;ap; Tran (1994， 225/32 d). W roundd th dcal part of th thr stpszs to 8 to ak th stpsz a ultpl of 2 n ordr to rduc th accuulaton of round-off rror n th coputaton procsss. In rlaton to ths， Wsdo ≈ap;ap;ap; Holan (1991) prford nurcal ntgratons of th outr fv plantary orts usng th syplctc ap th a stpsz of 400 d， 1/10.83 of th ortal prod of Juptr. Thr rsult ss to accurat nough， hch partly justfs our thod of dtrnng th stpsz. Hovr， snc th ccntrcty of Juptr (～0.05) s uch sallr than that of Mrcury (～0.2)， nd so car hn copar ths ntgratons sply n trs of stpszs.

In th ntgraton of th outr fv plants (F±)， fxd th stpsz at 400 d.

W adopt Gauss' f and g functons n th syplctc ap togthr th th thrd-ordr Hally thod (Dany 1992) as a solvr for Kplr quatons. Th nur of axu tratons st n Hally's thod s 15， ut thy nvr rachd th axu n any of our ntgratons.

Th ntrval of th data output s 200 000 d (～547 yr) for th calculatons of all nn plants (N±1，2，3)， and aout 8000 000 d (～21 903 yr) for th ntgraton of th outr fv plants (F±).

Although no output fltrng as don hn th nurcal ntgratons r n procss， appld a lo-pass fltr to th ra ortal data aftr had copltd all th calculatons. S Scton 4.1 for or dtal.

2.4 Error staton

2.4.1 Rlatv rrors n total nrgy and angular ontu

Accordng to on of th asc proprts of syplctc ntgrators， hch consrv th physcally consrvatv quantts ll (total ortal nrgy and angular ontu)， our long-tr nurcal ntgratons s to hav n prford th vry sall rrors. Th avragd rlatv rrors of total nrgy (～10?9) and of total angular ontu (～10?11) hav rand narly constant throughout th ntgraton prod (Fg. 1). Th spcal startup procdur， ar start， ould hav rducd th avragd rlatv rror n total nrgy y aout on ordr of agntud or or.

Rlatv nurcal rror of th total angular ontu δA/A0 and th total nrgy δE/E0 n our nurcal ntgratonsN± 1，2，3， hr δE and δA ar th asolut chang of th total nrgy and total angular ontu， rspctvly， andE0andA0ar thr ntal valus. Th horzontal unt s Gyr.

Not that dffrnt opratng systs， dffrnt athatcal lrars， and dffrnt hardar archtcturs rsult n dffrnt nurcal rrors， through th varatons n round-off rror handlng and nurcal algorths. In th uppr panl of Fg. 1， can rcognz ths stuaton n th scular nurcal rror n th total angular ontu， hch should rgorously prsrvd up to achn-ε prcson.

2.4.2 Error n plantary longtuds

Snc th syplctc aps prsrv total nrgy and total angular ontu of N-ody dynacal systs nhrntly ll， th dgr of thr prsrvaton ay not a good asur of th accuracy of nurcal ntgratons， spcally as a asur of th postonal rror of plants， .. th rror n plantary longtuds. To stat th nurcal rror n th plantary longtuds， prford th follong procdurs. W copard th rsult of our an long-tr ntgratons th so tst ntgratons， hch span uch shortr prods ut th uch hghr accuracy than th an ntgratons. For ths purpos， prford a uch or accurat ntgraton th a stpsz of 0.125 d (1/64 of th an ntgratons) spannng 3 × 105 yr， startng th th sa ntal condtons as n th N?1 ntgraton. W consdr that ths tst ntgraton provds us th a ‘psudo-tru’ soluton of plantary ortal voluton. Nxt， copar th tst ntgraton th th an ntgraton， N?1. For th prod of 3 × 105 yr， s a dffrnc n an anoals of th Earth tn th to ntgratons of ～0.52°(n th cas of th N?1 ntgraton). Ths dffrnc can xtrapolatd to th valu ～8700°， aout 25 rotatons of Earth aftr 5 Gyr， snc th rror of longtuds ncrass lnarly th t n th syplctc ap. Slarly， th longtud rror of Pluto can statd as ～12°. Ths valu for Pluto s uch ttr than th rsult n Knoshta ≈ap;ap;ap; Naka (1996) hr th dffrnc s statd as ～60°.

3 Nurcal rsults – I. Glanc at th ra data

In ths scton rfly rv th long-tr stalty of plantary ortal oton through so snapshots of ra nurcal data. Th ortal oton of plants ndcats long-tr stalty n all of our nurcal ntgratons: no ortal crossngs nor clos ncountrs tn any par of plants took plac.

3.1 Gnral dscrpton of th stalty of plantary orts

Frst， rfly look at th gnral charactr of th long-tr stalty of plantary orts. Our ntrst hr focuss partcularly on th nnr four trrstral plants for hch th ortal t-scals ar uch shortr than thos of th outr fv plants. As can s clarly fro th planar ortal confguratons shon n Fgs 2 and 3， ortal postons of th trrstral plants dffr lttl tn th ntal and fnal part of ach nurcal ntgraton， hch spans svral Gyr. Th sold lns dnotng th prsnt orts of th plants l alost thn th sar of dots vn n th fnal part of ntgratons () and (d). Ths ndcats that throughout th ntr ntgraton prod th alost rgular varatons of plantary ortal oton ran narly th sa as thy ar at prsnt.

Vrtcal v of th four nnr plantary orts (fro th z -axs drcton) at th ntal and fnal parts of th ntgratonsN±1. Th axs unts ar au. Th xy -plan s st to th nvarant plan of Solar syst total angular ontu.(a) Th ntal part ofN+1 ( t = 0 to 0.0547 × 10 9 yr).() Th fnal part ofN+1 ( t = 4.9339 × 10 8 to 4.9886 × 10 9 yr).(c) Th ntal part of N?1 (t= 0 to ?0.0547 × 109 yr).(d) Th fnal part ofN?1 ( t =?3.9180 × 10 9 to ?3.9727 × 10 9 yr). In ach panl， a total of 23 684 ponts ar plottd th an ntrval of aout 2190 yr ovr 5.47 × 107 yr . Sold lns n ach panl dnot th prsnt orts of th four trrstral plants (takn fro DE245).