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1( 1( o d¤–L (Hfi ˘U'XHfi210093)` ? NN [ Hermite {§¿|^T {O 1(3 1( ; z „§ X ›‰ 1( UN z .· XNN .(N=7)§=rY(!7(!/¥ \ § q (§ 3 1( NCO\ UN§X z m •c ›‰5 ⁄\UN m ’X§Œ (Jw«U ›‰ 3u 1( UN ? 1025kg’ cUN ˘ Œ {§NN [;1(X X1 g1801cPiazziuy1 1(Ceres–5§ 8k LŒ 1( uy§ O 1( oŒ Lz 1( »k1000œp m(X 1(Ceres)§ . 1(K k ˛ ,⁄k 1( O (16§2)£10¡10M¯[6]( 1(Ceres ¥ n' ı) ıŒ 1( 8¥3»( 7(; m§ k95 1(; »32.17)3.64U' m§ø m« ¡ 1( § u p 1(¡ / 1(0U; Œ O' §; 35 NC %˙0.17NC? 1(Œ8 ıKirkwood Y·U 1(; » O ˝¶ ' A ˜ 1( ·X /⁄ ” 8Evk ‘{1770c mJ Titius-Bode{K@ §1(; e^U' 5O §dS 1( F/† ldn(AU) –de † œ“L dn=0:4+0:3£2n¡2;n=¡1;2;3;4;¢¢¢T{KU’ (/L« 1( l ’XUdœ“§3n=5 / 1(§ 53d?uy 1( Øu 1( § k «w{ 1( ¿‘ ⁄‹‘[11] 1( ¿‘·dOlbersJ §ƒ@ ø 1(· ¿ 1(⁄3e { ¡ ˜ u— d˘‘§@ 1( U·, 1(†Lıg ¿ /⁄ ⁄‹‘K@ § 1(Ø U· /⁄ 1( c 3 X/⁄—ˇ § 1( NC N D 7( ˜§ƒ1(/⁄L§˚33(f ª§ z 8U 1( ¤E-mail:peijunlee@yahoo.com.cn ,yfiuy 1(Œ8flı§ · ¥ 1(Ceres k8:7£1020kg§= u/¥ ' ı• ˜ flK·3 1( 1(·˜ –›‰ 3§XJ –@o ?·ı ” d§• ^NNŒ ¨' {§ ˜ 1( b 1(3 ƒ 1( ^e z „• ˜k31 !¥0 'ƒ^ Œ {))Hermite {§1n!¥0 • ^ . {§1o!0 1( b 1(›‰5 Œ (J§ ' ( 2Hermite {Hermite {· «o $ {§AO· U ‹ 5O 2.1 {£ªb kN :, t0 ¥ 'O ~xi(t0);~vi(t0)(i=1¢¢¢N)Øu ¥, :~x=~x0=~x(t0);~v=~v0=~v(t0), ⁄ \ ~a0, \ m Œ _~a0~a0=XjGmj~rjr3j;_~a0=XjGmj_~rjr3j¡3(_~rj¢~rj)~rjr5j# ¥~rj=~x¡~xj;_~rj=~v¡~vj;rj=j~rjj;j6=iØu$ Cqk:~vp(t)=12(t1¡t0)2_~a0+(t1¡t0)~a0+~v0~xp(t)=16(t1¡t0)3_~a0+12(t1¡t0)2~a0+(t1¡t0)~v0+~x0ˇL “ –rt1 \ ~a1 \ m Œ_~a1(‰e5, ¡§~a1 _~a1 –ˇLt0 V—m“ _~a1=12(t1¡t0)2~a(3)0+(t1¡t0)~a(2)0+_~a0~a1=16(t1¡t0)3~a(3)0+12(t1¡t0)2~a(2)0+(t1¡t0)_~a0+~a0dd –) ~a(2)0 ~a(3)0:16~a(3)0=2~a0¡~a1(t1¡t0)3+_~a0+_~a1(t1¡t0)212~a(2)0=¡3~a0¡~a1(t1¡t0)2¡2_~a0+_~a1t1¡t0|^ “ –rt1 ¥ —m p /“ ~v(t1)=~vp(t)+16(t1¡t0)3~a(2)0+124(t1¡t0)4~a(3)0~x(t1)=~xp(t)+124(t1¡t0)4~a(2)0+1120(t1¡t0)5~a(3)0ø ·o Hermite {§§· « ) { ªHermite { {z/“ [8] ~v(t1)=~v0+12(~a0+~a1)¢t¡112(_~a1¡_~a0)¢t2(1)~x(t1)=~x0+12(~v0+~v1)¢t¡110(~a1¡~a0)¢t2+1120(_~a1+_~a0)¢t3(2) ¥¢t=t1¡t0§(2)“ /“[2][5] ~x(t1)=~x0+12(~v0+~v1)¢t¡112(~a1¡~a0)¢t2(3)(1)“ ·O(¢t5)§(2)“ ·O(¢t6)§(1) (2) O(¢t4);(3)“ ·O(¢t5)(1)!(2) (3)/“eU O(¢t4)d(1)!(2) (3)“ /“ –w Hermite {· mØ¡ 2.2 m Aarseth[1] O † œ“ ¢t=s´j~ajj~a(2)j+j_~aj2j_~ajj~a(3)j+j~a(2)j2(4) –ˇL´5 ,¢SA^¥†~ ´ 30.01)0.04 mIO m œ“(4)3NN [¥kX2 A^§Sweatman[10]Ød ı ’ ? n @ Hermite { –2 A^ug X ¥ l1(X z »˘ [,–e ? Ø1(X Hermite {3 m N z¥‹ \X § , –ˇL 5 § ·ø ’ ⁄O mØu ˇ~k « {§ «· ‰ §, « { ·ˇL† œ“O fl¢ XJ ^ ‰ § » ‹k ˇCz[7]Kokubo,Yoshinaga&Makino[4]J ıg Hermite {(P(EC)n;n=1;2;3;4)§l –k Jp 2.3 N /Ø N$˜§ ^ıg Hermite {P(EC)n§(Jw« ‰ ,X oU vk ˇCzª1w« N$˜¥ » Ø %˙ Ø Cz§ z m'O 20 105 ; –ˇØuHermite {P(EC)n(n=3;4)§ » Ø –9 %˙ Ø vk ˇCz§ Øu(n=1;2) k†w Jpª2w« %˙eU Ø Cz§ m ‰§z –ˇ·200 dª §ıg U w˝/Jp » %˙ § gŒ ı§ Jp †w¶ ·d { gŒn¸3 §‰ˆ{? Jp §¿ %˙ u0.2 ¥ ŒO (P(EC)2) ˜ ’(PEC)$4 Œ ?§ø · {(EC)⁄Jp Øu ƒ ; Œ§ E, 3 ˇ 5Cz=2,3,4n=1-4-3-2-10102468101214161820Δe/e0x10-7OrbitNumbern=2,3,4n=1-2-1.5-1-0.500.5012345678910Δa/a0x10-7OrbitNumberx104n=3,4n=2-10-8-6-4-20012345678910Δe/e0x10-8OrbitNumberx104n=3,4n=2ª1: N$˜¥ » Ø –9 %˙ Ø m Cz§ gŒ n;e=0:1§¨' 200 /–ˇ » Ø Cz,m : %˙ Cz§¨' m 20 ; –ˇ; e: » Ø Cz,me: %˙ Cz§¨' m 105 ; –ˇ;Figure1:Therelativeerrorsoforbitelementsa;eoftheP(EC)nHermiteschemesagainsttime.Theintegrationstepsizeis200stepsperorbit.TopLeft:Therelativeerrorsofa,TopRight:Therelativeerrorsofe,theintegraltimeis20orbitalperiods;BottomLeft:Therelativeerrorsofa,BottomRight:Therelativeerrorsofe,theintegraltimeis105orbitalperiods.|ΔE/E0|lg(e)n=1n=2n=3n=4ª2:U Ø %˙ Cz(ØŒª)105 ; –ˇ§ gŒ n§¨' 200 /–ˇFigure2:Therelativeenergyerrorafter105orbitalperiodsfortheP(EC)nHermiteschemesagainsttheeccentricityinKeplerproblem(logpicture),theKeplerperiodis2¼intimeunits.Theintegrationstepsizeis200stepsperorbit3 . {aqLaskar[9] {§ Y(!7(!/¥ \ § q (§ 3»( 7( mO\ UN§? z „edb 1( ; »(‰ 7( ; ) ,K TX ›‰§ K ›‰ §=›‰; »a %˙e v^ aMars(1+eMars)a(1¡e)·a(1+e)aJupiter(1¡eJupiter)¨' Jo Hermite {§¨' m •cfi XUN —' ^DE405(JED2440400.5)⁄ Œ §b 1( ; »a %˙e — ¯ £a 32:0AU¡4:0AU m§e ·0¡0:3⁄§ 20Ø(=20^; ) §; Œ ; i=0±,, :† =0±,C: !=0±,—'†C: M0=0Øb 1( M=2:0£10jkg(j=20;20:5;21;21:5;22;24;25;26)'O ˜ ; ›‰54Œ (J ^Hermite {P(EC)3 ¨' §• ˜ ªX ˜ ˘ z§¨' ‰( »(; –ˇ 1=200)ª3 ¥ « /ØA —' » %˙' § ªØAb 1( 2£1021kg§mªØAb 1( 2£1025kgª¥ ^› b 1( ; »(‰ 7(; u) '.› L1·b 1( ØA ›‰X ⁄ ’~§XJr ØA ›‰’~ ª§K ª4dª § Xb 1( O\§›‰