tonths， wsdonbsp;& holn 1991 perford nurcl tegrtonsthe outer fve ry orbts usg the sylectttepsze400 d， 110.83the orbtl peroesult seebe urte enough， whch prtly justfes our thoddeterng th， sce the entrctyjupter 0.05ch sller thn thtrcury 0.2，needcre when we pre these tegrtons slyterstepszes.
the tegrtonthe outer fve pls f，fxed the stepsze400 d.
we dopt guss&039; f nunctonsthe sylectother wth the thrdorder hlley thod dnby 1992 solver for keple nuerxbsp;tertonssethlley&039;s thod15， but they never reched the xbsp; nr tegrtons.
the tervlthe dt output200 000 d 547for the clctonsll ne pls n1，2，3， nd bout 8000 000 d903for the tegrtonthe outer fve pls f.
lthoughoutput flterg ws done when the nurcl tegrtons wereprocess，ppleowpss flterthe rw orbtl dt fterhd pleted llsecton 4.1 fordetl.
2.4 error estton
2.4.1 rtve errorstotl energy nd ngr ntubr>
ordgohe bsc propertessylectc tegrtors， whch conserve the physclly conservtve qunttes well totl orbtl energy nd ngr ntu our longternbsp;nurcl tegrtons seenbsp;to hve been perford wth veryverd rtve errorstotl energy 109 ndtotl ngr ntunbsp;1011 hve reed nerly constnt throughout the tegrton pero specl strtup procedure， wrnbsp;strt， would hve reduced the verd rtve errortotl energybout one ordergntudere.
rtve nurcl errorthe totl ngr ntunbsp;δ0 nd the totl energy δee0our nurcl tegrtonsn 1，2，3， wherendre the bsolute chnthe totl energy nd totl ngr ntu respectvely， nde0nd0re ther t horzontl untgyr.
note tht dfferent opertg syste， dfferent thetcl lbrres， nd dfferent hrdwre rchtectures resultdfferent nurcl errors， through the vrtonsroundoff error hndlg nd nurc the upper pnel o，cn recognze ths stutonthe secr nurcl errorthe totl ngr ntu whch shouldrgorously preservedto chee precson.
2.4.2 errorry longtudes
sce the sylectcpreserve totl energy nd totl ngr ntunbsp;of nbody dyncl syste herently well， the degreether preservtoot good surethe urcynurcl tegrtons， especlly surethe postonl errorpls， .e. the errorr estte the nurcl errorthe ry longtudes，perford the follow pred the resultourlongternbsp;tegrtons wthtest tegrtons， whch spnshorter perods but wthhgher urcy thn thhs purpose，perford re urte tegrton wth d 164thetegrtons spnng 3105 yr， strtg wth thetl condtons thonsder tht ths test tegrton provdeswtseudotrue solutonry orbt， we pre the test tegrton wth thetegrton， n1. for the perod3105 yr，sefferencen nolesthe erth between the two the csethfference cnextrptedthe vlue 8700， boutrottonserth fter 5 gyr， sce the errorlongtudes creses lerly wth the sylectc p. srly， the longtude errorpluto cnestted12. ths vlue for plutoch better thn the resultkosht & nk 1996 where the dfferenceestted60.
3 nurcl results – . ncethe rw dt
ths sectonbrefly revew the longternbsp;stbltyry orbtl ton throughsnpshotsrw nurc orbtl tonpls dctes longternbsp;stbltyllour nurcl tegrtons:orbtl crossgs nor close encounters between ny prpls took ce.
