Th varaton of ccntrcts and ortal nclnatons for th nnr four plants n th ntal and fnal part of th ntgraton N+1 s shon n Fg. 4. As xpctd， th charactr of th varaton of plantary ortal lnts dos not dffr sgnfcantly tn th ntal and fnal part of ach ntgraton， at last for Vnus， Earth and Mars. Th lnts of Mrcury， spcally ts ccntrcty， s to chang to a sgnfcant xtnt. Ths s partly caus th ortal t-scal of th plant s th shortst of all th plants， hch lads to a or rapd ortal voluton than othr plants; th nnrost plant ay narst to nstalty. Ths rsult appars to n so agrnt th Laskar's (1994， 1996) xpctatons that larg and rrgular varatons appar n th ccntrcts and nclnatons of Mrcury on a t-scal of svral 109 yr. Hovr， th ffct of th possl nstalty of th ort of Mrcury ay not fatally affct th gloal stalty of th hol plantary syst ong to th sall ass of Mrcury. W ll nton rfly th long-tr ortal voluton of Mrcury latr n Scton 4 usng lo-pass fltrd ortal lnts.

Th ortal oton of th outr fv plants ss rgorously stal and qut rgular ovr ths t-span (s also Scton 5).

3.2 T–frquncy aps

Although th plantary oton xhts vry long-tr stalty dfnd as th non-xstnc of clos ncountr vnts， th chaotc natur of plantary dynacs can chang th oscllatory prod and apltud of plantary ortal oton gradually ovr such long t-spans. Evn such slght fluctuatons of ortal varaton n th frquncy doan， partcularly n th cas of Earth， can potntally hav a sgnfcant ffct on ts surfac clat syst through solar nsolaton varaton (cf. Brgr 1988).

To gv an ovrv of th long-tr chang n prodcty n plantary ortal oton， prford any fast Fourr transforatons (FFTs) along th t axs， and suprposd th rsultng prodgras to dra to-dnsonal t–frquncy aps. Th spcfc approach to drang ths t–frquncy aps n ths papr s vry spl – uch splr than th avlt analyss or Laskar's (1990， 1993) frquncy analyss.

Dvd th lo-pass fltrd ortal data nto any fragnts of th sa lngth. Th lngth of ach data sgnt should a ultpl of 2 n ordr to apply th FFT.

Each fragnt of th data has a larg ovrlappng part: for xapl， hn th th data gns fro t=t and nds at t=t+T， th nxt data sgnt rangs fro t+δT≤t+δT+T， hr δT?T. W contnu ths dvson untl rach a crtan nur N y hch tn+T rachs th total ntgraton lngth.

W apply an FFT to ach of th data fragnts， and otan n frquncy dagras.

In ach frquncy dagra otand aov， th strngth of prodcty can rplacd y a gry-scal (or colour) chart.

W prfor th rplacnt， and connct all th gry-scal (or colour) charts nto on graph for ach ntgraton. Th horzontal axs of ths n graphs should th t， .. th startng ts of ach fragnt of data (t， hr = 1，…， n). Th vrtcal axs rprsnts th prod (or frquncy) of th oscllaton of ortal lnts.

W hav adoptd an FFT caus of ts ovrhlng spd， snc th aount of nurcal data to dcoposd nto frquncy coponnts s trrly hug (svral tns of Gyts).

A typcal xapl of th t–frquncy ap cratd y th aov procdurs s shon n a gry-scal dagra as Fg. 5， hch shos th varaton of prodcty n th ccntrcty and nclnaton of Earth n N+2 ntgraton. In Fg. 5， th dark ara shos that at th t ndcatd y th valu on th ascssa， th prodcty ndcatd y th ordnat s strongr than n th lghtr ara around t. W can rcognz fro ths ap that th prodcty of th ccntrcty and nclnaton of Earth only changs slghtly ovr th ntr prod covrd y th N+2 ntgraton. Ths narly rgular trnd s qualtatvly th sa n othr ntgratons and for othr plants， although typcal frquncs dffr plant y plant and lnt y lnt.

4.2 Long-tr xchang of ortal nrgy and angular ontu

W calculat vry long-prodc varaton and xchang of plantary ortal nrgy and angular ontu usng fltrd Dlaunay lnts L， G， H. G and H ar quvalnt to th plantary ortal angular ontu and ts vrtcal coponnt pr unt ass. L s rlatd to th plantary ortal nrgy E pr unt ass as E=?μ2/2L2. If th syst s copltly lnar， th ortal nrgy and th angular ontu n ach frquncy n ust constant. Non-lnarty n th plantary syst can caus an xchang of nrgy and angular ontu n th frquncy doan. Th apltud of th lost-frquncy oscllaton should ncras f th syst s unstal and raks don gradually. Hovr， such a sypto of nstalty s not pronnt n our long-tr ntgratons.

In Fg. 7， th total ortal nrgy and angular ontu of th four nnr plants and all nn plants ar shon for ntgraton N+2. Th uppr thr panls sho th long-prodc varaton of total nrgy (dnotd asE- E0)， total angular ontu ( G- G0)， and th vrtcal coponnt ( H- H0) of th nnr four plants calculatd fro th lo-pass fltrd Dlaunay lnts.E0， G0， H0 dnot th ntal valus of ach quantty. Th asolut dffrnc fro th ntal valus s plottd n th panls. Th lor thr panls n ach fgur shoE-E0，G-G0 andH-H0 of th total of nn plants. Th fluctuaton shon n th lor panls s vrtually ntrly a rsult of th assv jovan plants.

Coparng th varatons of nrgy and angular ontu of th nnr four plants and all nn plants， t s apparnt that th apltuds of thos of th nnr plants ar uch sallr than thos of all nn plants: th apltuds of th outr fv plants ar uch largr than thos of th nnr plants. Ths dos not an that th nnr trrstral plantary susyst s or stal than th outr on: ths s sply a rsult of th rlatv sallnss of th asss of th four trrstral plants copard th thos of th outr jovan plants. Anothr thng notc s that th nnr plantary susyst ay co unstal or rapdly than th outr on caus of ts shortr ortal t-scals. Ths can sn n th panls dnotd asnnr 4 n Fg. 7 hr th longr-prodc and rrgular oscllatons ar or apparnt than n th panls dnotd astotal 9. Actually， th fluctuatons n thnnr 4 panls ar to a larg xtnt as a rsult of th ortal varaton of th Mrcury. Hovr， cannot nglct th contruton fro othr trrstral plants， as ll s n susqunt sctons.

4.4 Long-tr couplng of svral nghourng plant pars

Lt us s so ndvdual varatons of plantary ortal nrgy and angular ontu xprssd y th lo-pass fltrd Dlaunay lnts. Fgs 10 and 11 sho long-tr voluton of th ortal nrgy of ach plant and th angular ontu n N+1 and N?2 ntgratons. W notc that so plants for apparnt pars n trs of ortal nrgy and angular ontu xchang. In partcular， Vnus and Earth ak a typcal par. In th fgurs， thy sho ngatv corrlatons n xchang of nrgy and postv corrlatons n xchang of angular ontu. Th ngatv corrlaton n xchang of ortal nrgy ans that th to plants for a closd dynacal syst n trs of th ortal nrgy. Th postv corrlaton n xchang of angular ontu ans that th to plants ar sultanously undr crtan long-tr prturatons. Canddats for prturrs ar Juptr and Saturn. Also n Fg. 11， can s that Mars shos a postv corrlaton n th angular ontu varaton to th Vnus–Earth syst. Mrcury xhts crtan ngatv corrlatons n th angular ontu vrsus th Vnus–Earth syst， hch ss to a racton causd y th consrvaton of angular ontu n th trrstral plantary susyst.

It s not clar at th ont hy th Vnus–Earth par xhts a ngatv corrlaton n nrgy xchang and a postv corrlaton n angular ontu xchang. W ay possly xplan ths through osrvng th gnral fact that thr ar no scular trs n plantary sajor axs up to scond-ordr prturaton thors (cf. Brour ≈ap;ap;ap; Clnc 1961; Boccaltt ≈ap;ap;ap; Pucacco 1998). Ths ans that th plantary ortal nrgy (hch s drctly rlatd to th sajor axs a) ght uch lss affctd y prturng plants than s th angular ontu xchang (hch rlats to ). Hnc， th ccntrcts of Vnus and Earth can dsturd asly y Juptr and Saturn， hch rsults n a postv corrlaton n th angular ontu xchang. On th othr hand， th sajor axs of Vnus and Earth ar lss lkly to dsturd y th jovan plants. Thus th nrgy xchang ay ltd only thn th Vnus–Earth par， hch rsults n a ngatv corrlaton n th xchang of ortal nrgy n th par.

As for th outr jovan plantary susyst， Juptr–Saturn and Uranus–Nptun s to ak dynacal pars. Hovr， th strngth of thr couplng s not as strong copard th that of th Vnus–Earth par.

5 ± 5 × 1010-yr ntgratons of outr plantary orts

Snc th jovan plantary asss ar uch largr than th trrstral plantary asss， trat th jovan plantary syst as an ndpndnt plantary syst n trs of th study of ts dynacal stalty. Hnc， addd a coupl of tral ntgratons that span ± 5 × 1010 yr， ncludng only th outr fv plants (th four jovan plants plus Pluto). Th rsults xht th rgorous stalty of th outr plantary syst ovr ths long t-span. Ortal confguratons (Fg. 12)， and varaton of ccntrcts and nclnatons (Fg. 13) sho ths vry long-tr stalty of th outr fv plants n oth th t and th frquncy doans. Although do not sho aps hr， th typcal frquncy of th ortal oscllaton of Pluto and th othr outr plants s alost constant durng ths vry long-tr ntgraton prods， hch s donstratd n th t–frquncy aps on our pag.

In ths to ntgratons， th rlatv nurcal rror n th total nrgy as ～10?6 and that of th total angular ontu as ～10?10.

5.1 Rsonancs n th Nptun–Pluto syst

Knoshta ≈ap;ap;ap; Naka (1996) ntgratd th outr fv plantary orts ovr ± 5.5 × 109 yr . Thy found that four ajor rsonancs tn Nptun and Pluto ar antand durng th hol ntgraton prod， and that th rsonancs ay th an causs of th stalty of th ort of Pluto. Th ajor four rsonancs found n prvous rsarch ar as follos. In th follong dscrpton，λ dnots th an longtud，Ω s th longtud of th ascndng nod and ? s th longtud of prhlon. Suscrpts P and N dnot Pluto and Nptun.

Man oton rsonanc tn Nptun and Pluto (3:2). Th crtcal argunt θ1= 3 λP? 2 λN??P lrats around 180° th an apltud of aout 80° and a lraton prod of aout 2 × 104 yr.

Th argunt of prhlon of Pluto ωP=θ2=?P?ΩP lrats around 90° th a prod of aout 3.8 × 106 yr. Th donant prodc varatons of th ccntrcty and nclnaton of Pluto ar synchronzd th th lraton of ts argunt of prhlon. Ths s antcpatd n th scular prturaton thory constructd y Koza (1962).

Th longtud of th nod of Pluto rfrrd to th longtud of th nod of Nptun，θ3=ΩP?ΩN， crculats and th prod of ths crculaton s qual to th prod of θ2 lraton. Whn θ3 cos zro， .. th longtuds of ascndng nods of Nptun and Pluto ovrlap， th nclnaton of Pluto cos axu， th ccntrcty cos nu and th argunt of prhlon cos 90°. Whn θ3 cos 180°， th nclnaton of Pluto cos nu， th ccntrcty cos axu and th argunt of prhlon cos 90° agan. Wllas ≈ap;ap;ap; Bnson (1971) antcpatd ths typ of rsonanc， latr confrd y Mlan， Nol ≈ap;ap;ap; Carpno (1989).

An argunt θ4=?P??N+ 3 (ΩP?ΩN) lrats around 180° th a long prod，～ 5.7 × 108 yr.

In our nurcal ntgratons， th rsonancs ()–() ar ll antand， and varaton of th crtcal argunts θ1，θ2，θ3 ran slar durng th hol ntgraton prod (Fgs 14–16 ). Hovr， th fourth rsonanc (v) appars to dffrnt: th crtcal argunt θ4 altrnats lraton and crculaton ovr a 1010-yr t-scal (Fg. 17). Ths s an ntrstng fact that Knoshta ≈ap;ap;ap; Naka's (1995， 1996) shortr ntgratons r not al to dsclos.

6 Dscusson

What knd of dynacal chans antans ths long-tr stalty of th plantary syst? W can datly thnk of to ajor faturs that ay rsponsl for th long-tr stalty. Frst， thr s to no sgnfcant lor-ordr rsonancs (an oton and scular) tn any par aong th nn plants. Juptr and Saturn ar clos to a 5:2 an oton rsonanc (th faous ‘grat nqualty’)， ut not just n th rsonanc zon. Hghr-ordr rsonancs ay caus th chaotc natur of th plantary dynacal oton， ut thy ar not so strong as to dstroy th stal plantary oton thn th lft of th ral Solar syst. Th scond fatur， hch thnk s or portant for th long-tr stalty of our plantary syst， s th dffrnc n dynacal dstanc tn trrstral and jovan plantary susysts (Ito ≈ap;ap;ap; Tankaa 1999， 2001). Whn asur plantary sparatons y th utual Hll rad (R_)， sparatons aong trrstral plants ar gratr than 26RH， hras thos aong jovan plants ar lss than 14RH. Ths dffrnc s drctly rlatd to th dffrnc tn dynacal faturs of trrstral and jovan plants. Trrstral plants hav sallr asss， shortr ortal prods and dr dynacal sparaton. Thy ar strongly prturd y jovan plants that hav largr asss， longr ortal prods and narror dynacal sparaton. Jovan plants ar not prturd y any othr assv ods.

Th prsnt trrstral plantary syst s stll ng dsturd y th assv jovan plants. Hovr， th d sparaton and utual ntracton aong th trrstral plants rndrs th dsturanc nffctv; th dgr of dsturanc y jovan plants s O(J)(ordr of agntud of th ccntrcty of Juptr)， snc th dsturanc causd y jovan plants s a forcd oscllaton havng an apltud of O(J). Hghtnng of ccntrcty， for xapl O(J)～0.05， s far fro suffcnt to provok nstalty n th trrstral plants havng such a d sparaton as 26RH. Thus assu that th prsnt d dynacal sparaton aong trrstral plants (≈ap;ap;gt; 26RH) s proaly on of th ost sgnfcant condtons for antanng th stalty of th plantary syst ovr a 109-yr t-span. Our dtald analyss of th rlatonshp tn dynacal dstanc tn plants and th nstalty t-scal of Solar syst plantary oton s no on-gong.

Although our nurcal ntgratons span th lft of th Solar syst， th nur of ntgratons s far fro suffcnt to fll th ntal phas spac. It s ncssary to prfor or and or nurcal ntgratons to confr and xan n dtal th long-tr stalty of our plantary dynacs.

——以上文段引自 Ito， T.≈ap;ap; Tankaa， K. Long-tr ntgratons and stalty of plantary orts n our Solar Syst. Mon. 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（《Natur》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。