{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "2D Math" -1 2 "Times" 1 11 0 0 0 1 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 0 0 128 128 1 2 0 1 0 0 0 0 0 0 } {CSTYLE "2D Comment" 2 18 "" 0 0 0 128 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "2D Input" 2 19 "" 0 0 128 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "2D Output " 2 20 "" 0 0 0 0 128 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Non-proportional" -1 256 "Courier" 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Symbol" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Times" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 261 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 262 "" 1 13 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 18 263 "" 1 13 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 0 2 0 0 0 0 0 0 }0 1 0 -1 6 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading \+ 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 24 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 15 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 6 4 0 0 0 0 0 0 -1 0 } {PSTYLE "Heading 3" 4 5 1 {CSTYLE "" -1 -1 "" 1 13 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 2 2 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Fixed Width" 0 17 1 {CSTYLE "" -1 -1 "Co urier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 17 0 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "Times" 1 22 0 128 128 1 0 1 0 0 0 0 0 0 0 }3 1 0 -1 14 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "Times" 1 16 0 0 128 1 0 1 0 0 0 0 0 0 0 }3 1 0 -1 8 0 0 0 0 0 0 0 261 0 }{PSTYLE "Heading 4" 5 20 1 {CSTYLE "" -1 -1 "" 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 }0 0 0 -1 2 0 0 0 0 0 0 0 -1 0 } {PSTYLE "Address" 0 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 128 1 1 2 0 0 0 0 0 0 0 }3 1 0 -1 6 -1 0 0 0 0 0 0 263 0 }{PSTYLE "Date" 0 257 1 {CSTYLE "" -1 -1 "" 1 14 0 0 128 1 2 0 0 0 0 0 0 0 0 }3 0 0 -1 8 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Address - email" -1 258 1 {CSTYLE "" -1 -1 "Verdana" 1 10 0 0 128 1 2 0 0 0 0 0 0 0 0 }3 0 0 -1 4 -1 0 0 0 0 0 0 262 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "Trebuchet MS" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "Trebuchet MS" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "Treb uchet MS" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 263 1 {CSTYLE "" -1 -1 "Trebuchet MS" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 264 1 {CSTYLE "" -1 -1 "Trebuchet MS" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 265 1 {CSTYLE "" -1 -1 "Treb uchet MS" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 266 1 {CSTYLE "" -1 -1 "Trebuchet MS" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 18 "" 0 "" {TEXT -1 102 "Characterization of Approximate Solutions\nof Kepler's Equation\nand\nOptimization of Time Series Sto rage" }}{PARA 19 "" 0 "" {TEXT -1 15 "Marc A. Murison" }}{PARA 256 "" 0 "" {TEXT -1 74 "Astronomical Applications Department\nU.S. Naval Obs ervatory\nWashington, DC" }}{PARA 258 "" 0 "" {TEXT -1 57 "murison@aa. usno.navy.mil\nhttp://aa.usno.navy.mil/murison/" }}{PARA 257 "" 0 "" {TEXT -1 13 "June 19, 1998" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Cont ents " }}{PARA 259 "" 0 "" {HYPERLNK 17 "Introduction" 1 "" "Introduct ion" }}{PARA 260 "" 0 "" {HYPERLNK 17 "Expansion in eccentricity" 1 " " "Expansion in eccentricity" }}{PARA 261 "" 0 "" {HYPERLNK 17 "Expans ion in time" 1 "" "Expansion in time" }}{PARA 266 "" 0 "" {HYPERLNK 17 "" 1 "" "Error properties of different polynomial expansions" } {HYPERLNK 17 "Characterizing the error term for a given time span" 1 " " "Characterizing the error term for a given time span" }}{PARA 263 " " 0 "" {HYPERLNK 17 "Optimization of expression size" 1 "" "Optimizati on of expression size" }}{PARA 264 "" 0 "" {HYPERLNK 17 "Miscellaneous 1: y as a function of e and M" 1 "" "Miscellaneous 1: y as a function of e and M" }}{PARA 265 "" 0 "" {HYPERLNK 17 "Miscellaneous 2: a diff erent approximation approach" 1 "" "Miscellaneous 2: a different appro ximation approach" }}}{SECT 1 {PARA 3 "" 0 "Introduction" {TEXT -1 12 "Introduction" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Kepler's equation can be written " }{XPPEDIT 18 0 "f(phi) = 0;" "6#/-%\"fG6#%$phiG\"\"! " }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "f(phi) = phi-e*sin(phi)-M:" " 6#/-%\"fG6#%$phiG,(F'\"\"\"*&%\"eGF)-%$sinG6#F'F)!\"\"%\"MGF/" }{TEXT -1 12 ", and where " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 27 " i s the eccentric anomaly, " }{XPPEDIT 18 0 "M = n*t;" "6#/%\"MG*&%\"nG \"\"\"%\"tGF'" }{TEXT -1 22 " is the mean anomaly, " }{TEXT 259 1 "n" }{TEXT -1 25 " is the mean motion, and " }{TEXT 257 1 "e" }{TEXT -1 35 " is the orbital eccentricity. Let " }{TEXT 258 1 "x" }{TEXT -1 43 " be an approximation for the true solution " }{XPPEDIT 18 0 "phi; " "6#%$phiG" }{TEXT -1 10 ", and let " }{XPPEDIT 18 0 "delta;" "6#%&de ltaG" }{TEXT -1 28 " be the error in the guess: " }{XPPEDIT 18 0 "phi \+ = xi+delta;" "6#/%$phiG,&%#xiG\"\"\"%&deltaGF'" }{TEXT -1 21 ". Then w e can expand " }{XPPEDIT 18 0 "f(xi+delta);" "6#-%\"fG6#,&%#xiG\"\"\"% &deltaGF(" }{TEXT -1 57 " in a Taylor series. For example, to third o rder we have" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "restart;" "6#%( restartG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "with(DEtools):" "6# -%%withG6#%(DEtoolsG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "f(phi) \+ = phi-e*sin(phi)-M:" "6#/-%\"fG6#%$phiG,(F'\"\"\"*&%\"eGF)-%$sinG6#F'F )!\"\"%\"MGF/" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Kepler := fn(r hs(%),phi,e,M);" "6#>%'KeplerG-%#fnG6&-%$rhsG6#%\"%G%$phiG%\"eG%\"MG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'KeplerGR6%%$phiG%\"eG%\"MG6\"6$%) operatorG%&arrowGF*,(9$\"\"\"*&9%F0-%$sinG6#F/F0!\"\"9&F6F*F*F*" }}} {EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "f(phi) = convert(expansion(f(xi+ delta),delta,3),diff);" "6#/-%\"fG6#%$phiG-%(convertG6$-%*expansionG6% -F%6#,&%#xiG\"\"\"%&deltaGF2F3\"\"$%%diffG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%$phiG,*-F%6#%#xiG\"\"\"*&-%%diffG6$F)F+F,%&de ltaGF,F,*&-F/6$F)-%\"$G6$F+\"\"#F,)F1F8\"\"\"#F,F8*&-F/6$F)-F66$F+\"\" $F,)F1FAF:#F,\"\"'" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "ODE := %: " "6#>%$ODEG%\"%G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "This is equi valent to the differential operator form " }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "f(phi) = de2diffop(rhs(ODE),f(xi),[Dx, xi])*f(xi);" "6# /-%\"fG6#%$phiG*&-%*de2diffopG6%-%$rhsG6#%$ODEG-F%6#%#xiG7$%#DxGF2\"\" \"-F%6#F2F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%$phiG*&-%*de2 diffopG6%,*-F%6#%#xiG\"\"\"*&-%%diffG6$F-F/F0%&deltaGF0F0*&-F36$F--%\" $G6$F/\"\"#F0)F5F<\"\"\"#F0F<*&-F36$F--F:6$F/\"\"$F0)F5FEF>#F0\"\"'F-7 $%#DxGF/F0F-F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "where the argum ent in parenthesis is a differential operator with " }{XPPEDIT 18 0 "D x*f = diff(f,xi);" "6#/*&%#DxG\"\"\"%\"fGF&-%%diffG6$F'%#xiG" }{TEXT -1 3 ". " }}}}{SECT 1 {PARA 3 "" 0 "Expansion in eccentricity" {TEXT -1 25 "Expansion in eccentricity" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "Take the old-fashioned approach and expand the solution in a polyn omial in eccentricity." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "trial soln := proc(phi,e,C,n::\{posint,name\})\n local k;\n phi = sum(C[k] *e^k,k = 0 .. n);\nend:" "6#>%*trialsolnGR6&%$phiG%\"eG%\"CG'%\"nG<$%' posintG%%nameG7#%\"kG6\"F1/F'-%$sumG6$*&&F)6#F0\"\"\")F(F0F9/F0;\"\"!F +F1F1F1" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Ne := 6:" "6#>%#NeG \"\"'" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "trialsoln(phi,e,C,Ne); " "6#-%*trialsolnG6&%$phiG%\"eG%\"CG%#NeG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "trial := %:" "6#>%&trialG%\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/%$phiG,0&%\"CG6#\"\"!\"\"\"*&&F'6#F*F*%\"eGF*F**&&F'6#\"\"#F*)F.F2 \"\"\"F**&&F'6#\"\"$F*)F.F8F4F**&&F'6#\"\"%F*)F.F=F4F**&&F'6#\"\"&F*)F .FBF4F**&&F'6#\"\"'F*)F.FGF4F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Stick this back into the Kepler equation, then expand on eccentricity to get" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(trial,Kepler(ph i,e,M) = 0):" "6#-%%subsG6$%&trialG/-%'KeplerG6%%$phiG%\"eG%\"MG\"\"! " }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "expansion(%,e,Ne);" "6#-%*expansion G6%%\"%G%\"eG%#NeG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "eqnx := %:" "6#>% %eqnxG%\"%G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,2*&,(*&-%$sinG6#&%\"C G6#\"\"!\"\"\",(*&&F,6#\"\"#F/)&F,6#F/\"\"$\"\"\"#F/\"\"'*&&F,6#\"\"%F /F6F/!\"\"*&&F,6#F8F/F2F9F@F/F@*&-%$cosGF*F/,**$)F6\"\"&F9#F/\"$?\"*&) F2F4F9F6F9#F@F4&F,6#FJF/*&FBF9)F6F4F9FOF/F@&F,6#F;F/F/)%\"eGF;F9F/*&,( FPF/*&FEF9,&*&F2F9FSF9FOF=F/F/F@*&F(F9,(*&FBF9F6F9F@*$)F6F?F9#F/\"#C*$ FNF9FOF/F@F/)FWFJF9F/*&,(F=F/*(F(F9F2F9F6F9F/*&FEF9,&FBF/*$F5F9#F@F;F/ F@F/)FWF?F9F/*&,(*&F(F9FSF9#F/F4*&FEF9F2F9F@FBF/F/)FWF8F9F/*&,&*&FEF9F 6F9F@F2F/F/)FWF4F9F/*&,&F6F/F(F@F/FWF/F/F+F/%\"MGF@F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Solve successively for the coefficients. \+ We'll write a quickie procedure to do this automatically." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "xsolve := proc( expr::algebraic, var::n ame, coef::name )\n local k, kmax, eqn, sols;\n kmax := degree( coll ect(expr,var), var );\n sols := [];\n for k from 0 to kmax do\n e qn := coeff( expr, var, k );\n sols := [ op(sols), isolate( subs(so ls,eqn), coef[k] ) ];\n od;\n sols;\nend:" "6#>%'xsolveGR6%'%%exprG% *algebraicG'%$varG%%nameG'%%coefGF,7&%\"kG%%kmaxG%$eqnG%%solsG6\"F4C&> F1-%'degreeG6$-%(collectG6$F(F+F+>F37\"?(F0\"\"!\"\"\"F1%%trueGC$>F2-% &coeffG6%F(F+F0>F37$-%#opG6#F3-%(isolateG6$-%%subsG6$F3F2&F.6#F0F3F4F4 F4" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "xsolve(lhs(eqnx),e,C);" " 6#-%'xsolveG6%-%$lhsG6#%%eqnxG%\"eG%\"CG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "trial_coeffs := %:" "6#>%-trial_coeffsG%\"%G" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#7)/&%\"CG6#\"\"!%\"MG/&F&6#\"\"\"-%$sinG6#F)/&F&6#\" \"#*&-%$cosGF0F-F.F-/&F&6#\"\"$,&*$)F.F;\"\"\"#!\"\"F4*&)F6F4F?F.F?F-/ &F&6#\"\"%,&*&F>F?F6F?FA*&F6F?,&F=#!\"#F;FBF-F-F-/&F&6#\"\"&,&*&F6F?,& FI#!\"$F4FJF-F-F-*&F.F?,(*&FF?F@FSF-FWF-*&F%%solnG%\"% G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%$phiG,0%\"MG\"\"\"*&-%$sinG6#F& F'%\"eGF'F'*(-%$cosGF+F'F)\"\"\")F,\"\"#F0F'*&,&*$)F)\"\"$F0#!\"\"F2*& )F.F2F0F)F0F'F')F,F7F0F'*&,&*&F6F0F.F0F9*&F.F0,&F5#!\"#F7F:F'F'F'F')F, \"\"%F0F'*&,&*&F.F0,&F?#!\"$F2F@F'F'F'*&F)F0,(*&F4F0F)F0F9*$)F)FEF0#F' \"#C*&F;F0)F)F2F0F8F'F'F')F,\"\"&F0F'*&,&*&F)F0,(*&F.F0FPF0#F'\"\"'*&F >F0F)F0F9*(F4F0F.F0F)F0F9F'F'*&F.F0,,*$)F)FVF0#F'\"$?\"*&F;F0F6F0F8FHF 'FLF'*&F4F0FTF0F8F'F'F')F,FgnF0F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Plot just the coefficients." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 4 "plot" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "plot([0, seq(rhs(tria l_coeffs[k]),k = 2 .. 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" }{TEXT -1 14 ". Substitute " }{XPPEDIT 18 0 "M = M[0]+delta;" "6#/% \"MG,&&F$6#\"\"!\"\"\"%&deltaGF)" }{TEXT -1 15 " and expand on " } {XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 66 ". For illustration, we look at just the second-order expansion. " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "long expression" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "expansion(subs(M = M[0]+delta,soln),delta,2);" "6#-%*expansionG6 %-%%subsG6$/%\"MG,&&F*6#\"\"!\"\"\"%&deltaGF/%%solnGF0\"\"#" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%$phiG,4*&,.*&-%$sinG6#&%\"MG6#\"\"!\"\"\"% \"eGF0#!\"\"\"\"#*(-%$cosGF+F0F)\"\"\")F1F4F8!\"#*&,**&F)F8,&*$)F)F4F8 F3*$)F6F4F8F0F0F2*$)F)\"\"$F8#F0\"\"%*&FBF8F)F8#!\"(F4*&,&FAF3F?F0F0F) F8F0F0)F1FEF8F0*&,,*&FDF8F6F8#\"\"(F4*&,(F=F0FCF2FHF4F0F6F8F3*&F6F8,*F =#F:FEFC#F0FEFH#!#B\"\"'FKF0F0F0*&F6F8,&FCFWFHF0F0F2*&F)F8,&*&F@F8F6F8 !\"%*$)F6FEF8F0F0F3F0)F1FGF8F0*&,.*&F6F8,,FP#\"#@FGFS#!\"$F4FUF0FfnF2F hnF3F0F0*&F6F8,&FPFeoFfnF0F0F2*&F)F8,**$)F)FGF8#FEF4*&FBF8F@F8#!\"*F4* &F6F8FinF8F0*&F)F8FgnF8F3F0F3*&F)F8,0*&,&FCF2FHF0F0F)F8#F0F4*&FF8#F0\"#7F^p#\"#8Fen*&FBF8F>F8F2*&FLF 8F@F8F2F0F0*&F)F8,(FepF3F[p#F0\"#CF^pF2F0F2*&F6F8,**&FfpF8F6F8F3*&FjpF 8F)F8F3FP#FRFen*&F]oF8F)F8F3F0F0F0)F1\"\"&F8F0*&,.*&F)F8,2*&F6F8,&F[qF 4F^pFGF0#F0Fen*&F6F8F\\pF8#FfoFG*&,&FPF3FfnF0F0F)F8Fgp*&FOF8F)F8F3*&,* F[pF0F^pFfoFapF0FbpF3F0F6F8F3*(FfpF8F6F8F)F8Fgp*&,(FhqF2*&F*&F)F8FcrF8#F0\"$?\"Fhs#F3\"$S#Ffs#\"#\"* \"#I*&FBF8FTF8F2*&FLF8FDF8F2FaoF0FgoF2FioF3FcpF0FbqF2FfqF0*&FfpF8F>F8F 2*&FF8F9F8F0*&FjpF8FMF8F0F0F0*&F[sF8F^oF8F0*&,* FfuF0FguF3FhuF0FiuF0F0F\\rF8F0*&,**&F)F8FesF8F0*&F6F8FcsF8F0*&F6F8FbuF 8F0*&F)F8F_uF8F3F0F[vF8F0F0F]vF0F0F,F0*&FhrF8F^oF8F0F(F0*&,&FgoF0FbqF0 F0F\\rF8F0F5F0*&,&FbsF0F^uF0F0F[vF8F0*&FfpF8FMF8F0" }}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "cost(%);" "6#-%%costG6#%\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%*additionsG\"$r$%0multiplicationsG\"%+8%*functi onsG\"$U(%,assignmentsG\"\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 196 "As we can see, expansions of useful order will generate large express ions. However, the appearance of these expressions can be considerabl y simplified by converting to Fourier form. For example," }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "collect(combine(%%,trig),[delta, e]);" "6#-%(collectG6$-%(combineG6$%#%%G%%trigG7$%&deltaG%\"eG" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#/%$phiG,4*&,.*&,(-%$sinG6#,$&%\"MG6#\"\"!\"\"##! \"\"\"#C-F+6#,$F.\"\"'#!$V#\"#S-F+6#,$F.\"\"%#\"#K\"#:\"\"\")%\"eGF9\" \"\"FD*&,(-F+6#,$F.\"\"&#!%DJ\"$o(-F+6#,$F.\"\"$#\"$V#\"$c#-F+6#F.#F4 \"$%QFD)FFFMFGFD*&,&F*#FDFTF=#!\")FTFD)FFF@FGFD*&,&FQ#!#F\"#;FX#FDFaoF D)FFFTFGFD*&)FFF2FGF*FDF4*&FXFDFFFD#F4F2FD)%&deltaGF2FGFD*&,0*&,(-%$co sGF>#!#;FC-F_pF7#\"#\")F<-F_pF,#FDF5FDFEFGFD*&,(-F_pFR#!#\")\"$G\"-F_p FK#\"$D'Fen-F_pFY#FD\"$#>FDFfnFGFD*&,&F^p#F@FTFep#F4FTFDF\\oFGFD*&,&F` q#F4\"\")Fip#\"\"*FjqFDFcoFGFD*&FepFDFeoFGFD*&F`qFDFFFGFDFDFDFDFioFDFD *&,(F*#FD\"#[F=#!\"%FCF6#\"#F\"#!)FDFEFGFD*&,(FJ#\"$D\"FenFQ#F`oF\\qFX FaqFDFfnFGFD*&,&F=FinF*#F4F9FDF\\oFGFD*&,&FQ#FTFjqFXFiqFDFcoFGFDFdo#FD F2FfoFDF.FD" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "cost(%);" "6#-%% costG6#%\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%*additionsG\"#P%0mu ltiplicationsG\"$D\"%*functionsG\"#O%,assignmentsG\"\"\"" }}}}{SECT 1 {PARA 3 "" 0 "Characterizing the error term for a given time span" {TEXT -1 51 "Characterizing the error term for a given time span" }} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Introduction" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 40 "Suppose we have a specified error bound " } {XPPEDIT 18 0 "eta;" "6#%$etaG" }{TEXT -1 65 ". We would like to say \+ something about the approximation orders " }{XPPEDIT 18 0 "N[e];" "6#& %\"NG6#%\"eG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "N[t];" "6#&%\"NG6#% \"tG" }{TEXT -1 27 " required to get the error " }{XPPEDIT 18 0 "epsil on;" "6#%(epsilonG" }{TEXT -1 11 " less than " }{XPPEDIT 18 0 "eta;" " 6#%$etaG" }{TEXT -1 36 " as a function of the time interval " } {XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT -1 29 ". That is, find N such th at " }{XPPEDIT 18 0 "epsilon(N[e],N[t]) < eta;" "6#2-%(epsilonG6$&%\"N G6#%\"eG&F(6#%\"tG%$etaG" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "First, we write a function which returns an approximatio n of the solution of the Kepler equation, incorporating expansions in \+ both eccentricity and time." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 " KeplerApprox := proc( phi::name, e::name, M::name, \n \+ mu::name, Ne::posint, NM::posint )\n local C, tmp, cpu;\n cpu : = time();\n trialsoln(phi,e,C,Ne);\n expansion( subs(%,Kepler(phi,e, M)), e, Ne );\n xsolve( %, e, C );\n subs(%,%%%);\n tmp := expansio n( subs(M=M[0]+mu,%), mu, NM );\n if assigned(verbosity) and verbosit y > 0 then\n print( `cost(raw)`=cost(tmp) );\n tmp := collect( c ombine(tmp,trig), [mu,e] );\n print( `cost(Fourier)`=cost(tmp) );\n print( `series expansion cpu`=time() - cpu );\n tmp;\n else\n \+ collect( combine(tmp,trig), [mu,e] );\n fi;\nend:" "6#>%-KeplerApp roxGR6('%$phiG%%nameG'%\"eGF)'%\"MGF)'%#muGF)'%#NeG%'posintG'%#NMGF27% %\"CG%$tmpG%$cpuG6\"F9C)>F8-%%timeGF9-%*trialsolnG6&F(F+F6F1-%*expansi onG6%-%%subsG6$%\"%G-%'KeplerG6%F(F+F-F+F1-%'xsolveG6%FGF+F6-FE6$FG%$% %%G>F7-FB6%-FE6$/F-,&&F-6#\"\"!\"\"\"F/FenFGF/F4@%3-%)assignedG6#%*ver bosityG2FZF[oC'-%&printG6#/%*cost(raw)G-%%costG6#F7>F7-%(collectG6$-%( combineG6$F7%%trigG7$F/F+-F_o6#/%.cost(Fourier)G-Fdo6#F7-F_o6#/%5serie s~expansion~cpuG,&-F=F9FenF8!\"\"F7-Fho6$-F[p6$F7F]p7$F/F+F9F9F9" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "For example, " }{XPPEDIT 19 1 "Kep lerApprox(phi,e,M,mu,5,3);" "6#-%-KeplerApproxG6(%$phiG%\"eG%\"MG%#muG \"\"&\"\"$" }{TEXT -1 7 " yields" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/% $phiG,4*&,,*&,(-%$cosG6#,$&%\"MG6#\"\"!\"\"$#\"$V#\"$c#-F+6#,$F.\"\"&# !&Dc\"\"%/B-F+6#F.#!\"\"\"%_6\"\"\")%\"eGF9\"\"\"FB*&,&-F+6#,$F.\"\"## FK\"\"*-F+6#,$F.\"\"%#!#KFMFB)FDFQFEFB*&,&F*#!#F\"#;F=#FB\"#[FB)FDF2FE FB*&FHFB)FDFKFE#!\"#F2*&F=FBFDFB#F@\"\"'FB)%#muGF2FEFB*&,,*&,(-%$sinGF ,F3-FeoF7#!%DJ\"$o(-FeoF>#F@\"$%QFBFCFEFB*&,&-FeoFO#!\")F2-FeoFI#FBF2F BFTFEFB*&,&FdoFWFjo#FBFYFBFfnFEFB*&FhnFEFbpFBF@*&FjoFBFDFE#F@FKFB)F_oF KFEFB*&,.*&,(F*#!#\")\"$G\"F6#\"$D'F\\pF=#FB\"$#>FBFCFEFB*&,&FH#F@F2FN #FQF2FBFTFEFB*&,&F=#F@\"\")F*#FMF]rFBFfnFEFBFgnFBF[oFBFBFBFBF_oFBFB*&, (Ffo#\"$D\"F\\pFdo#FXFaqFjoFdqFBFCFEFB*&,&F_pFcpFbpF\\oFBFTFEFB*&,&Fdo #F2F]rFjoF\\rFBFfnFEFBFgp#FBFKFhpFBF.FB" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Here is a procedure to extract the highest-order term." } }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "error_term := proc( e::name, M ::name, mu::name, \n Ne::posint, NM::posint )\n lo cal phi;\n KeplerApprox(phi,e,M,mu,Ne,NM);\n lcoeff(rhs(%),mu)*mu^de gree(rhs(%),mu);\nend:" "6#>%+error_termGR6''%\"eG%%nameG'%\"MGF)'%#mu GF)'%#NeG%'posintG'%#NMGF07#%$phiG6\"F5C$-%-KeplerApproxG6(F4F(F+F-F/F 2*&-%'lcoeffG6$-%$rhsG6#%\"%GF-\"\"\")F--%'degreeG6$-F?6#FAF-FBF5F5F5 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "For example, " }{XPPEDIT 19 1 "error_term(e,M,mu,5,3);" "6#-%+error_termG6'%\"eG%\"MG%#muG\"\"&\" \"$" }{TEXT -1 7 " yields" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&,,*&,(- %$cosG6#,$&%\"MG6#\"\"!\"\"&#!&Dc\"\"%/B-F(6#F+#!\"\"\"%_6-F(6#,$F+\" \"$#\"$V#\"$c#\"\"\")%\"eGF/\"\"\"F?*&,&-F(6#,$F+\"\"##FH\"\"*-F(6#,$F +\"\"%#!#KFJF?)FAFNFBF?*&,&F8#!#F\"#;F3#F?\"#[F?)FAF;FBF?*&FEF?)FAFHFB #!\"#F;*&F3F?FAF?#F6\"\"'F?)%#muGF;FB" }}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "plot" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "Plot the coeff icient of " }{XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "G := fn(subs(M[0] = M,lcoeff(%,mu)),e,M );" "6#>%\"GG-%#fnG6%-%%subsG6$/&%\"MG6#\"\"!F--%'lcoeffG6$%\"%G%#muG% \"eGF-" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"GGR6$%\"eG%\"MG6\"6$%)op eratorG%&arrowGF),,*&,(-%$cosG6#,$9%\"\"$#\"$V#\"$c#-F16#,$F4\"\"&#!&D c\"\"%/B-F16#F4#!\"\"\"%_6\"\"\")9$F<\"\"\"FE*&,&-F16#,$F4\"\"##FN\"\" *-F16#,$F4\"\"%#!#KFPFE)FGFTFHFE*&,&F0#!#F\"#;F@#FE\"#[FE)FGF5FHFE*&FK FE)FGFNFH#!\"#F5*&F@FEFGFE#FC\"\"'F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "p0 := plot3d(0, M = 0 .. 360, e = 0 .. .3, style = patc hnogrid, color = black):" "6#>%#p0G-%'plot3dG6'\"\"!/%\"MG;F(\"$g$/%\" eG;F($\"\"$!\"\"/%&styleG%,patchnogridG/%&colorG%&blackG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "p1 := plot3d(G(e, M*deg),M = 0 .. 360,e = 0 .. .3,style = patchcontour,contours = 35,grid = [50, 50]):" "6#>% #p1G-%'plot3dG6(-%\"GG6$%\"eG*&%\"MG\"\"\"%$degGF./F-;\"\"!\"$g$/F+;F2 $\"\"$!\"\"/%&styleG%-patchcontourG/%)contoursG\"#N/%%gridG7$\"#]\"#] " }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "plots[display3d]([p0, p1],l abels = [\"M\", \"e\", \"\"],orientation = [125, 65],title = \"coeff(5 ,3)\");" "6#-&%&plotsG6#%*display3dG6&7$%#p0G%#p1G/%'labelsG7%Q\"M6\"Q \"eF0Q!F0/%,orientationG7$\"$D\"\"#l/%&titleGQ+coeff(5,3)6\"" }}{PARA 13 "" 1 "" {GLPLOT3D 611 424 424 {PLOTDATA 3 "6.-%%GRIDG6';\"\"!$\"$g$ F';F'$\"\"$!\"\"W(\\bm\":\":000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000000000 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%*AXESSTYLEG6#%$BOXG-%+LIGHTMODELG6#%(LIGHT_4G-%)CONTOURSG6#\"#N-%(SCA LINGG6#%.UNCONSTRAINEDG-%*AXESTICKSG6&%(DEFAULTGFhnFhn-FG6$FI\"#7-%+AX ESLABELSG6&Q\"MFEQ\"eFEQ!6\"FF-F46#%&PATCHG-%+PROJECTIONG6%$\"$D\"F'$ \"#lF'$\"\")F-" 1 2 0 1 0 2 4 6 2 2 0.800000 65.000000 125.000000 2 }} }}}}{SECT 1 {PARA 4 "" 0 "The coefficients of mu^k" {TEXT -1 20 "The c oefficients of " }{XPPEDIT 261 0 "mu^k;" "6#)%#muG%\"kG" }{TEXT -1 1 " " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 12 "Introduction" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "The double series expansion (in eccentric ity and in mean anomaly or time) is of the form " }{XPPEDIT 18 0 "phi \+ = sum(sum(C[k,n](M[0])*e^n,n = 0 .. N[e])*mu^k,k = 0 .. N[t]);" "6#/%$ phiG-%$sumG6$*&-F&6$*&-&%\"CG6$%\"kG%\"nG6#&%\"MG6#\"\"!\"\"\")%\"eGF1 F7/F1;F6&%\"NG6#F9F7)%#muGF0F7/F0;F6&F=6#%\"tG" }{TEXT -1 10 ". Defin e " }{XPPEDIT 18 0 "F[k,N[e]](e,M[0]) = sum(C[k,n](M[0])*e^n,n = 0 .. \+ N[e]);" "6#/-&%\"FG6$%\"kG&%\"NG6#%\"eG6$F,&%\"MG6#\"\"!-%$sumG6$*&-&% \"CG6$F(%\"nG6#&F/6#F1\"\"\")F,F:F>/F:;F1&F*6#F," }{TEXT -1 9 " so tha t " }{XPPEDIT 18 0 "phi = sum(F[k,N[e]](e,M[0])*mu^n,k = 0 .. N[t]);" "6#/%$phiG-%$sumG6$*&-&%\"FG6$%\"kG&%\"NG6#%\"eG6$F1&%\"MG6#\"\"!\"\" \")%#muG%\"nGF7/F-;F6&F/6#%\"tG" }{TEXT -1 17 ". The functions " } {XPPEDIT 18 0 "F[k,N[e]](e,M[0]);" "6#-&%\"FG6$%\"kG&%\"NG6#%\"eG6$F+& %\"MG6#\"\"!" }{TEXT -1 74 " will come up later, so we will characteri ze their typical behavior now. " }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "F := proc( e, M, Ne::posint, n::posint )\n local mu;\n error_t erm(e,M,mu,Ne,n);\n coeff(%,mu,n);\nend:" "6#>%\"FGR6&%\"eG%\"MG'%#Ne G%'posintG'%\"nGF+7#%#muG6\"F0C$-%+error_termG6'F'F(F/F*F--%&coeffG6%% \"%GF/F-F0F0F0" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 13 "Variation of \+ " }{XPPEDIT 262 0 "N[t];" "6#&%\"NG6#%\"tG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "F(e,M,3,1);" "6#-%\"FG6&%\"eG%\"MG\"\"$ \"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&,&-%$cosG6#&%\"MG6#\"\"! #!\"\"\"\")-F'6#,$F)\"\"$#\"\"*F/\"\"\")%\"eGF3\"\"\"F6*&-F'6#,$F)\"\" #F6)F8F>F9F6*&F&F6F8F6F6F6F6" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "F(e,M,3,2);" "6#-%\"FG6&%\"eG%\"MG\"\"$\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&-%$sinG6#,$&%\"MG6#\"\"!\"\"$#!#F\"#;-F'6#F*#\"\" \"F1F5)%\"eGF.\"\"\"F5*&)F7\"\"#F8-F'6#,$F*F;F5!\"\"*&F2F5F7F5#F?F;" } }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "F(e,M,3,3);" "6#-%\"FG6&%\"eG% \"MG\"\"$\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&-%$cosG6#,$&% \"MG6#\"\"!\"\"$#!#F\"#;-F'6#F*#\"\"\"\"#[F5)%\"eGF.\"\"\"F5*&-F'6#,$F *\"\"#F5)F8F>F9#!\"#F.*&F2F5F8F5#!\"\"\"\"'" }}}{EXCHG {PARA 0 "" 0 " " {XPPEDIT 19 1 "F(e,M,3,4);" "6#-%\"FG6&%\"eG%\"MG\"\"$\"\"%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&-%$sinG6#,$&%\"MG6#\"\"!\"\"$#\" #\")\"#k-F'6#F*#!\"\"\"$#>\"\"\")%\"eGF.\"\"\"F7*&)F9\"\"#F:-F'6#,$F*F =F7#F7F.*&F2F7F9F7#F7\"#C" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "F( e,M,3,5);" "6#-%\"FG6&%\"eG%\"MG\"\"$\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&-%$cosG6#&%\"MG6#\"\"!#!\"\"\"$g*-F'6#,$F)\"\"$# \"$V#\"$?$\"\"\")%\"eGF3\"\"\"F7*&-F'6#,$F)\"\"#F7)F9F?F:#F?\"#:*&F&F7 F9F7#F7\"$?\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "F(e,M,3,6);" " 6#-%\"FG6&%\"eG%\"MG\"\"$\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*& ,&-%$sinG6#,$&%\"MG6#\"\"!\"\"$#!$V#\"$S'-F'6#F*#\"\"\"\"%gdF5)%\"eGF. \"\"\"F5*&)F8\"\"#F9-F'6#,$F*F%'plot_FGR6%%%emaxG'%#NeG%'posintG'%\"nGF*7'%#FnG%\"MG%\"e 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