InSection2webrieflyexplainourdynamicalmodel， .Section4goesontoadiscussionofthelongest-termvariationofplanetaryorbitsusingalow-passfilterandincludesadiscussionofangularmomentumdeficit.InSection5， wepresentasetofnumericalintegrationsfortheouterfiveplanetsthatspans ± 5 × 1010yr.InSection6wealsodiscussthelong-termstabilityoftheplanetarymotionanditspossiblecause.
2Descriptionofthenumericalintegrations
（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。）
2.3Numericalmethod
Weutilizeasecond-orderWisdomCHolmansymplecticmapasourmainintegrationmethod (Wisdom & Holman1991; Kinoshita， Yoshida & Nakai1991) withaspecialstart-upproceduretoreducethetruncationerrorofanglevariables，‘warmstart’(Saha & Tremaine1992， 1994).
Thestepsizeforthenumericalintegrationsis8dthroughoutallintegrationsofthenineplanets (N±1，2，3)， whichisabout1/11oftheorbitalperiodoftheinnermostplanet (Mercury).Asforthedeterminationofstepsize， wepartlyfollowthepreviousnumericalintegrationofallnineplanetsinSussman & Wisdom (1988， 7.2d) andSaha & Tremaine (1994， 225/32d).， Wisdom & Holman (1991) performednumericalintegrationsoftheouterfiveplanetaryorbitsusingthesymplecticmapwithastepsizeof400d， 1/， ， sincetheeccentricityofJupiter (0.05) ismuchsmallerthanthatofMercury (0.2)， weneedsomecarewhenwecomparetheseintegrationssimplyintermsofstepsizes.
Intheintegrationoftheouterfiveplanets (F±)， wefixedthestepsizeat400d.
