.4 Errr esiain
.4.1 Relaive errrs in al energy and angular enu
Arding ne f he basi prperies f syplei inegrars， hih nserve he physially nservaive quaniies ell (al rbial energy and angular enu)， ur lng-er nuerial inegrains see have been perfred ih very sall errrs. he averaged relaive errrs f al energy (～10?9) and f al angular enu (～10?11) have reained nearly nsan hrughu he inegrain perid (Fig. 1). he speial sarup predure， ar sar， uld have redued he averaged relaive errr in al energy by abu ne rder f agniude r re.
Relaive nuerial errr f he al angular enu δA/A0 and he al energy δE/E0 in ur nuerial inegrainsN± 1，，， here δE and δA are he abslue hange f he al energy and al angular enu， respeively， andE0andA0are heir iniial values. he hriznal uni is Gyr.
Ne ha differen peraing syses， differen aheaial libraries， and differen hardare arhieures resul in differen nuerial errrs， hrugh he variains in rund-ff errr handling and nuerial algrihs. In he upper panel f Fig. 1， e an regnize his siuain in he seular nuerial errr in he al angular enu， hih shuld be rigrusly preserved up ahine-ε preisin.
.4. Errr in planeary lngiudes
Sine he syplei aps preserve al energy and al angular enu f N-bdy dynaial syses inherenly ell， he degree f heir preservain ay n be a gd easure f he auray f nuerial inegrains， espeially as a easure f he psiinal errr f planes， i.e. he errr in planeary lngiudes. esiae he nuerial errr in he planeary lngiudes， e perfred he flling predures. e pared he resul f ur ain lng-er inegrains ih se es inegrains， hih span uh shrer perids bu ih uh higher auray han he ain inegrains. Fr his purpse， e perfred a uh re aurae inegrain ih a sepsize f 0.15 d (1/64 f he ain inegrains) spanning × 1 ih he sae iniial ndiins as in he N?1 inegrain. e nsider ha his es inegrain prvides us ih a ‘pseud-rue’ sluin f planeary rbial evluin. Ne， e pare he es inegrain ih he ain inegrain， N?1. Fr he perid f × 105 yr， e see a differene in ean analies f he Earh beeen he inegrains f ～0.5°(in he ase f he N?1 inegrain). his differene an be eraplaed he value ～8700°， abu 5 rains f Earh afer 5 Gyr， sine he errr f lngiudes inreases linearly ih ie in he syplei ap. Siilarly， he lngiude errr f Plu an be esiaed as ～1°. his value fr Plu is uh beer han he resul in Kinshia ≈ap;ap; Nakai (1996) here he differene is esiaed as ～60°.
Nuerial resuls – I. Glane a he ra daa
In his sein e briefly revie he lng-er sabiliy f planeary rbial in hrugh se snapshs f ra nuerial daa. he rbial in f planes indiaes lng-er sabiliy in all f ur nuerial inegrains: n rbial rssings nr lse enuners beeen any pair f planes k plae.
.1 General desripin f he sabiliy f planeary rbis
Firs， e briefly lk a he general haraer f he lng-er sabiliy f planeary rbis. ur ineres here fuses pariularly n he inner fur erresrial planes fr hih he rbial ie-sales are uh shrer han hse f he uer five planes. As e an see learly fr he planar rbial nfigurains shn in Figs and ， rbial psiins f he erresrial planes differ lile beeen he iniial and final par f eah nuerial inegrain， hih spans several Gyr. he slid lines dening he presen rbis f he planes lie als ihin he sar f ds even in he final par f inegrains (b) and (d). his indiaes ha hrughu he enire inegrain perid he als regular variains f planeary rbial in reain nearly he sae as hey are a presen.
Verial vie f he fur inner planeary rbis (fr he z -ais direin) a he iniial and final pars f he inegrainsN±1. he aes unis are au. he y -plane is se he invarian plane f Slar syse al angular enu.(a) he iniial par fN+1 ( = 0 0.0547 × 10 9 yr).(b) he final par fN+1 ( = 4.99 × 10 8 4.9886 × 10 9 yr).() he iniial par f N?1 (= 0 ?0.0547 × 109 yr).(d) he final par fN?1 ( =?.9180 × 10 9 ?.977 × 10 9 yr). In eah panel， a al f 684 pins are pled ih an inerval f abu 190 yr ver 5.47 × 107 yr . Slid lines in eah panel dene he presen rbis f he fur erresrial planes (aken fr DE45).
he variain f eenriiies and rbial inlinains fr he inner fur planes in he iniial and final par f he inegrain N+1 is shn in Fig. 4. As epeed， he haraer f he variain f planeary rbial eleens des n differ signifianly beeen he iniial and final par f eah inegrain， a leas fr Venus， Earh and ars. he eleens f erury， espeially is eenriiy， see hange a signifian een. his is parly beause he rbial ie-sale f he plane is he shres f all he planes， hih leads a re rapid rbial evluin han her planes; he inners plane ay be neares insabiliy. his resul appears be in se agreeen ih Laskar's (1994， 1996) epeains ha large and irregular variains appear in he eenriiies and inlinains f erury n a ie-sale f several 109 yr. Hever， he effe f he pssible insabiliy f he rbi f erury ay n faally affe he glbal sabiliy f he hle planeary syse ing he sall ass f erury. e ill enin briefly he lng-er rbial evluin f erury laer in Sein 4 using l-pass filered rbial eleens.
he rbial in f he uer five planes sees rigrusly sable and quie regular ver his ie-span (see als Sein 5).
. ie–frequeny aps
Alhugh he planeary in ehibis very lng-er sabiliy defined as he nn-eisene f lse enuner evens， he hai naure f planeary dynais an hange he sillary perid and apliude f planeary rbial in gradually ver suh lng ie-spans. Even suh sligh fluuains f rbial variain in he frequeny dain， pariularly in he ase f Earh， an penially have a signifian effe n is surfae liae syse hrugh slar inslain variain (f. Berger 1988).
give an vervie f he lng-er hange in peridiiy in planeary rbial in， e perfred any fas Furier ransfrains (FFs) alng he ie ais， and superpsed he resuling peridgras dra -diensinal ie–frequeny aps. he speifi apprah draing hese ie–frequeny aps in his paper is very siple – uh sipler han he avele analysis r Laskar's (1990， 199) frequeny analysis.
Divide he l-pass filered rbial daa in any fragens f he sae lengh. he lengh f eah daa segen shuld be a uliple f in rder apply he FF.
Eah fragen f he daa has a large verlapping par: fr eaple， hen he ih daa begins fr =i and ends a =i+， he ne daa segen ranges fr i+δ≤i+δ+， here δ?. e ninue his divisin unil e reah a erain nuber N by hih n+ reahes he al inegrain lengh.
e apply an FF eah f he daa fragens， and bain n frequeny diagras.
In eah frequeny diagra bained abve， he srengh f peridiiy an be replaed by a grey-sale (r lur) har.
e perfr he replaeen， and nne all he grey-sale (r lur) hars in ne graph fr eah inegrain. he hriznal ais f hese ne graphs shuld be he ie， i.e. he saring ies f eah fragen f daa (i， here i= 1，…， n). he verial ais represens he perid (r frequeny) f he sillain f rbial eleens.
e have adped an FF beause f is verheling speed， sine he aun f nuerial daa be depsed in frequeny pnens is erribly huge (several ens f Gbyes).
A ypial eaple f he ie–frequeny ap reaed by he abve predures is shn in a grey-sale diagra as Fig. 5， hih shs he variain f peridiiy in he eenriiy and inlinain f Earh in N+ inegrain. In Fig. 5， he dark area shs ha a he ie indiaed by he value n he absissa， he peridiiy indiaed by he rdinae is srnger han in he ligher area arund i. e an regnize fr his ap ha he peridiiy f he eenriiy and inlinain f Earh nly hanges slighly ver he enire perid vered by he N+ inegrain. his nearly regular rend is qualiaively he sae in her inegrains and fr her planes， alhugh ypial frequenies differ plane by plane and eleen by eleen.
4. Lng-er ehange f rbial energy and angular enu
e alulae very lng-peridi variain and ehange f planeary rbial energy and angular enu using filered Delaunay eleens L， G， H. G and H are equivalen he planeary rbial angular enu and is verial pnen per uni ass. L is relaed he planeary rbial energy E per uni ass as E=?μ/L. If he syse is pleely linear， he rbial energy and he angular enu in eah frequeny bin us be nsan. Nn-lineariy in he planeary syse an ause an ehange f energy and angular enu in he frequeny dain. he apliude f he les-frequeny sillain shuld inrease if he syse is unsable and breaks dn gradually. Hever， suh a syp f insabiliy is n prinen in ur lng-er inegrains.
In Fig. 7， he al rbial energy and angular enu f he fur inner planes and all nine planes are shn fr inegrain N+. he upper hree panels sh he lng-peridi variain f al energy (dened asE- E0)， al angular enu ( G- G0)， and he verial pnen ( H- H0) f he inner fur planes alulaed fr he l-pass filered Delaunay eleens.E0， G0， H0 dene he iniial values f eah quaniy. he abslue differene fr he iniial values is pled in he panels. he ler hree panels in eah figure shE-E0，G-G0 andH-H0 f he al f nine planes. he fluuain shn in he ler panels is virually enirely a resul f he assive jvian planes.