{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple P lot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 78 "This worksheet is design ed to create plots of the trajectories of test masses " }}{PARA 256 " " 0 "" {TEXT -1 68 "orbiting in precessing elliptical orbits in the Sc hwarzschild metric" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name cha ngecoords has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Conservation of angular momentum:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Leq:=diff(phi(tau),tau)=B/r(tau)^ 2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$LeqG/-%%diffG6$-%$phiG6#%$tau GF,*&%\"BG\"\"\"-%\"rGF+!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "C onservation of energy:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Eeq:=diff(t(tau),tau) = A/(1-2*m/r(tau));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$EeqG/-%%diffG6$-%\"tG6#%$tauGF,*&% \"AG\"\"\",&F/F/*(\"\"#F/%\"mGF/-%\"rGF+!\"\"F6F6" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 16 "H:=1-2*m/r(tau);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"HG,&\"\"\"F&*(\"\"#F&%\"mGF&-%\"rG6#%$tauG!\"\"F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Hprime:=diff(1-2*m/r,r); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'HprimeG,$*(\"\"#\"\"\"%\"mGF(% \"rG!\"#F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Hprimeeq:=sub s(r=r(tau),Hprime);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)HprimeeqG,$* (\"\"#\"\"\"%\"mGF(-%\"rG6#%$tauG!\"#F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 "Those are the equations coming from the metric and the c onservation laws, i.e., time independence and spherical symmetry." }} {PARA 0 "" 0 "" {TEXT -1 91 " Of course they are using spherical sy mmetry to draw everything in the equatorial plane." }}{PARA 0 "" 0 "" {TEXT -1 50 " The equation below is the geodesic equation for r" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "r2eq:=diff(r(tau),tau$2)-B^2 *H/r(tau)^3+(1/2)*(Hprimeeq/H)*(A^2-diff(r(tau),tau)^2);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%%r2eqG,(-%%diffG6$-%\"rG6#%$tauG-%\"$G6$F,\"\" #\"\"\"*(%\"BGF0,&F1F1*(F0F1%\"mGF1F)!\"\"F7F1F)!\"$F7**F6F1F)!\"#F4F7 ,&*$)%\"AGF0F1F1*$)-F'6$F)F,F0F1F7F1F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 " Then I want to study the energy equations as well, whic h come from the normalizaton of the 4-velocity." }{MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Veff:=-(m/r(tau))+B^2/(2*r( tau)^2)-m*B^2/r(tau)^3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%VeffG,(* &%\"mG\"\"\"-%\"rG6#%$tauG!\"\"F-*&#F(\"\"#F(*&%\"BGF0F)!\"#F(F(*(F'F( F2F0F)!\"$F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "r1eq:=diff( r(tau),tau)^2/2 + Veff=(A^2-1)/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %%r1eqG/,**&#\"\"\"\"\"#F)*$)-%%diffG6$-%\"rG6#%$tauGF3F*F)F)F)*&%\"mG F)F0!\"\"F6*&F(F)*&%\"BGF*F0!\"#F)F)*(F5F)F9F*F0!\"$F6,&*&F*F6%\"AGF*F )#F)F*F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "ellipse:=subs( \{diff(r(tau),tau)=0\},r1eq);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(el lipseG/,(*&%\"mG\"\"\"-%\"rG6#%$tauG!\"\"F.*&#F)\"\"#F)*&%\"BGF1F*!\"# F)F)*(F(F)F3F1F*!\"$F.,&*&F1F.%\"AGF1F)#F)F1F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "Now let me just get rid of the mass by setting it t o the mass of the sun, as measured in kilometers:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 9 "m:=1.477;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG$ \"%x9!\"$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 " Next we state the values of the perihelion and aphelion distances of Mercury from the s un, again in kilometers:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "rpm:=46*10^6; ram:=69.82*10^6;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ rpmG\")+++Y" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ramG$\"++++#)p!\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "We introduce the speed of light here, in km/sec, but it may well not be of any use." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "c:=2.99792458*10^5;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG$\"+!eCz*H!\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "We may then insert those values in the e quation for the effective potential, as turn around points, " }}{PARA 0 "" 0 "" {TEXT -1 52 "to determine the associated values of the const ants." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ellperi:=subs(r(ta u)=rpm,ellipse);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(ellperiG/,&$\"+ l&p3@$!# " 0 "" {MPLTEXT 1 0 32 "ellap:=subs(r(tau)= ram,ellipse);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&ellapG/,&$\"+-(Ra6 #!# " 0 "" {MPLTEXT 1 0 37 "pars:=[solve(\{ellperi,el lap\},\{A,B\})];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%%parsG7&<$/%\"BG $!+_Jq]!*!\"'/%\"AG$!+s)*******!#5<$F'/F-$\"+s)*******F0<$F,/F($\"+_Jq ]!*F+<$F2F6" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Merc:=[op(pa rs[4])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%MercG7$/%\"AG$\"+s)**** ***!#5/%\"BG$\"+_Jq]!*!\"'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "ellipsewab:=subs(Merc,ellipse);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%+ellipsewabG/,(*&$\"%x9!\"$\"\"\"-%\"rG6#%$tauG!\"\"F0*&$\"+y8w&4%! \"#F+F,F4F+*&$\"+6z))47F0F+F,F*F0$!$G\"!#5" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 38 "testsol:=[solve(ellipsewab,\{r(tau)\})];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(testsolG7%<#/-%\"rG6#%$tauG$\"+:.+aH!\"*< #/F($\"+HE.MY!\"#<#/F($\"+w&H]!pF3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "Notice that these numbers ought to have been our original ones , namely .46 x 10^8 and .6982 x 10^8. They differ, already in the thi rd place?" }}{PARA 0 "" 0 "" {TEXT -1 37 " Let's see if we can do an y better?" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Digits:=25;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'DigitsG\"#D" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "ellperib:=subs(r(tau)=rpm, ellipse);ellapb:=subs(r(tau)=ram,ellipse);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%)ellperibG/,&$\":EyM/8R<_cp3@$!#K!\"\"*&$\":YR(4XR+c3 )[HO#!#S\"\"\")%\"BG\"\"#F/F/,&*&F2F*%\"AGF2F/#F/F2F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'ellapbG/,&$\":-(GyP\"4\"4-(Ra6#!#K!\"\"*&$\":]i !4\"yoqWCwc-\"!#S\"\"\")%\"BG\"\"#F/F/,&*&F2F*%\"AGF2F/#F/F2F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "parsb:=[solve(\{ellperib,ell apb\},\{A,B\})];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&parsbG7&<$/%\"A G$!:'4];)=`uC()*******!#D/%\"BG$!:y`SCrr.$\\Jq]!*!#@<$F,/F($\":'4];)=` uC()*******F+<$F'/F-$\":y`SCrr.$\\Jq]!*F0<$F2F6" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 16 "Mercb:=parsb[4];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&MercbG<$/%\"AG$\":'4];)=`uC()*******!#D/%\"BG$\":y`SCrr.$\\Jq ]!*!#@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ellipsewabb:=subs (Mercb,ellipse);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,ellipsewabbG/,( *&$\"%x9!\"$\"\"\"-%\"rG6#%$tauG!\"\"F0*&$\":52[j6\"3%[Phd4%!# " 0 "" {MPLTEXT 1 0 36 "testsb:=solve(ellipsewabb,\{r(tau)\});" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%'testsbG6%<#/-%\"rG6#%$tauG$\":EJ2M] QyYJ+S&H!#C<#/F($\":uzw6+++++++g%!#<<#/F($\":*)*))G(************>)pF3 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "AAAh Ha! Now, it's actually \+ doing the reasonable thing." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 137 "The first solution is then the correct values for t he constants of the motion for Mercury. We will insert them back int o the equations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 " However, first I'm going to try some numbers for the c onstants that are somewhat simpler, which correspond to a much closer \+ orbit. " }}{PARA 0 "" 0 "" {TEXT -1 64 "These will give us an orbit t hat \"precesses\" much more strongly." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "B:=3.5695; m:=1; A:=0.9509661236;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG$\"&&pN!\"%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG$\"+O7m4&*!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "Leq, r2eq;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%%diffG6$-%$phiG6#%$tauGF*,$*&$\"&&pN!\"%\"\"\"-%\"rG F)!\"#F0,(-F%6$F1-%\"$G6$F*\"\"#F0*($\"+DI8u7!\")F0,&F0F0*&F:F0F1!\"\" FAF0F1!\"$FA*(F1F3F?FA,&$\"5'pZ5[BolL/*!#?F0*$)-F%6$F1F*F:F0FAF0F0" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 " The three values below, then, also from the same book, tell me I'm going to start things up from re st at a turning point, where r(0)=11m." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "ini:=r(0)=11, D(r)(0)=0, phi(0)=0; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$iniG6%/-%\"rG6#\"\"!\"#6/--%\"D G6#F(F)F*/-%$phiGF)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "Eq 9:=dsolve(\{Leq,r2eq,ini\},\{r(tau),phi(tau)\},numeric,output=listproc edure);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$Eq9G7&/%$tauGf*6#F'6\"6# %aoCopyright~(c)~1993~by~the~University~of~Waterloo.~All~rights~reserv ed.GF*9$F*F*F*/-%$phiGF)f*F)6&%$resG%)solnprocG%)outpointG%&ndsolG6#%i nCopyright~(c)~2000~by~Waterloo~Maple~Inc.~All~rights~reserved.GF*C(>% 8_EnvDSNumericSaveDigitsG%'DigitsG>F<\"#D@%/%-_EnvInFsolveG%%trueG>8&- &%&evalfG6#F;6#F->FD-FGFI>8%f*6#%#mxG6+F5%#dtG%$datG%(odeprocG%#x0G%$v alG%$digG%\"nG%\"iG6#%inCopyright~(c)~2002~by~Waterloo~Maple~Inc.~All~ rights~reserved.GE\\s#Q0soln_proceduresF*=F*6#;\"\"\"\"\"%E\\[l%F[o\") KO#[\"\"\"#\")k,=9\"\"$\")/-=9F\\o\")+'oT\"Q(complexF*%&falseGC/>8$F-> FM=F*6#;F[oFaoE\\[l\"F[o=F*6#;F[o\"#9E\\[l/F[o6%/%)datatypeG%'sfloatG/ %&orderG%(C_orderG/%(storageG%,rectangularGF_oFcpFao7$\"\"!F^qF\\oX*%) anythingG6\"F*[gl%!%!!!\"8\"8!$!!!\"[^`r!\"!!\"!!!%!#:!\"!!\"\"&X*FfpF aqF*[gl!!%\"!!\"(\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 " The c ommand below, where I'm printing out the results of a plot command in \+ full, are mostly to show us how to create such a plot, and to " }} {PARA 0 "" 0 "" {TEXT -1 18 " interpret them." }{MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "PP1:= polarplot([rhs(Eq9(tau)[3]),rhs(Eq9(tau)[2]),tau=0..1],scaling=constra ined);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$PP1G-%/INTERFACE_PLOTG6%- %'CURVESG6$7S7$$\"#6\"\"!$F/F/7$$\":>!Gggj1lT(****4\"!#B$\":p'[HP)RTL( fb W'*Qd6FF=7$$\":y$)4A=Q`I,%***4\"F4$\":&=1?cdJ!yWm\\S$F=7$$\":w#>$>\\#[ %*Q:***4\"F4$\":OWKW9)z*ptGy/%F=7$$\":m_7rwBmt_))**4\"F4$\":\"o2%>JVuJ utMr%F=7$$\":47>1wdH9$\\)**4\"F4$\":\"\\k[bH\"G[p#)=S&F=7$$\":u2(en-r# *f3)**4\"F4$\":'4fuYb\"\\O_\"3)3'F=7$$\":PH9\\Y&G\\kh(**4\"F4$\":nm7F^ )yJp&=Rz'F=7$$\":R*>n4wia-;(**4\"F4$\":u#)[a2?&)fh #*)f'**4\"F4$\":y!QGzBh[x-]:\")F=7$$\":Xw_HO&3\"Q%)f**4\"F4$\":m;Pj%)R \"R%)HD=))F=7$$\":gn5O/.X*QM&**4\"F4$\":%ydouEir$e\"[&\\*F=7$$\":#Hyn] '3;B@Z**4\"F4$\":X&3gN*fm3lY5,\"!#D7$$\":SBSb8ER)*HR**4\"F4$\":;[;Sw7` !RQ<%3\"Fip7$$\":-)4r?sC![;K**4\"F4$\":Lz]\"\\Q')GO<7Y6Fip7$$\":?$z@y` &ejOB**4\"F4$\":%3a)zN2mxM\"==7Fip7$$\":,EW+=#)f28:**4\"F4$\":XEd)3o@l ?e'>G\"Fip7$$\":x0)Gm'*4b6c!**4\"F4$\":\"[%fX9d\\N$p%>N\"Fip7$$\":zab1 IlIo2'*)*4\"F4$\":U,t&*pNBe#\\e=9Fip7$$\":XdQrSK&QRc))*4\"F4$\":aW(e(z _k+09\")[\"Fip7$$\":\")\\sQwrDZhv)*4\"F4$\":vhA$4,1X(*G'>b\"Fip7$$\":2 fo8Qc#R=$3i\"Fip7$$\":?=0\"*==2!*3_)*4\"F4$\": U*>)4\"H'e2bmBp\"Fip7$$\":fNAG3c#4.5%)*4\"F4$\":8P7q_J:t1PYv\"Fip7$$\" :20a9B![3!eG)*4\"F4$\":k(yv!fL$e8-*=#=Fip7$$\":A*3U+VnAb_\")*4\"F4$\": L1')**4M(=#[o8*=Fip7$$\":\"e$oY97.pt,)*4\"F4$\":O*=+'R?#\\%eQ$f>Fip7$$ \":Njsmfn`H?)y*4\"F4$\":Aqy3C#e@XB5D?Fip7$$\":URdaGKqBls(*4\"F4$\":L`q T4So\\#37)4#Fip7$$\":Qt$p2;gm4#e(*4\"F4$\":n\"[@;h4;#GIP;#Fip7$$\":*)4 c\"\\OX5)HU(*4\"F4$\":/;Zk.]\\gT\"yLAFip7$$\":'))f4w*[P?WF(*4\"F4$\":. EnsD(H#Fip7$$\":l*)))zaj?Ts5(*4\"F4$\":%onV!3;*\\V@lmBFip7$$\":F :BC(GebRXp*4\"F4$\":&yIdri9\\Pj%>V#Fip7$$\":]'eV!**Hs6:x'*4\"F4$\":]9+ 8y4wFa'>+DFip7$$\":CK&QfMA'*4\"F4$\":/pU!Giuy JY0/FFip7$$\":*o\\QF.p1!)Gg*4\"F4$\":0-B$eSJu8U&Gx#Fip7$$\":\"4**eFnUM #4$e*4\"F4$\":%H:V#*fs)G)G3TGFip7$$\":+Hwuma%R&[k&*4\"F4$\":ma)*\\-g2# \\ox.HFip7$$\":byO6\\N=-mU&*4\"F4$\":&ytG$*fn]*))Hc(HFip7$$\":`)z_(\\J JvoA&*4\"F4$\":2_n[!3e]$)[*)RIFip7$$\":(QYc*zmnB#4]*4\"F4$\":.J$=%H'*y 4Z8%3JFip7$$\":EQQaOb6fjz%*4\"F4$\":@haO%*)G(z@$*R<$Fip7$$\"::8`sW\\__ Ec%*4\"F4$\":5#>tH\"*)G:=mWC$Fip-%'COLOURG6&%$RGBG$\"#5!\"\"F0F0-%(SCA LINGG6#%,CONSTRAINEDG-%%VIEWG6$%(DEFAULTGF]\\l" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 50 "nops(PP1), nops(op(1,PP1)), nops(op(1,op(1,PP1 )));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"$\"\"#\"#\\" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 145 "We interpret all the numbers just above \+ by saying that the dataset labeled as PP1 has 3 parts: the first part is the complete PLOT routine, the " }}{PARA 0 "" 0 "" {TEXT -1 134 "s econd part is the CONSTRAINED statement, which means that the vertical and horizontal axes should use the same scaling, and the third" }} {PARA 0 "" 0 "" {TEXT -1 56 "is a statement about where the plot shoul d be displayed." }}{PARA 0 "" 0 "" {TEXT -1 131 " Then the plot stat ement itself has two parts: they are the description of the CURVE to \+ be plotted, and then the colours to use." }}{PARA 0 "" 0 "" {TEXT -1 123 " Lastly the CURVE has some 49 points, which Maple thought was suf ficient to generate the desired more-or-less smooth curve." }}{PARA 0 "" 0 "" {TEXT -1 136 " Note that each of the 49 points is an x-coor d, and a y-coord, which the \"polarplot\" command generated from the v alues of r and theta" }}{PARA 0 "" 0 "" {TEXT -1 35 " which \+ were passed to it." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 " I also show below what such a plot actually looks like ." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "pol arplot([rhs(Eq9(tau)[3]),rhs(Eq9(tau)[2]),tau=0..300],scaling=constrai ned);" }}{PARA 13 "" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6%-%'CURV ESG6$7jq7$$\"#6\"\"!$F*F*7$$\":T%)oLB8O[7!>%4\"!#B$\":-Ttz)QOmPw5f5!#C 7$$\":!=5bt\"e-5['zw5F/$\":@MCB?=#*f+%*p5#F27$$\":QnvL$HTU!y?B0\"F/$\" :z)f].'*>8O6(3+$F27$$\":/*38s*>$>WWF>5F/$\":Zq/n&z(=8w#HqQF27$$\":H()p /:W@Log2s*F2$\":$R&H!y[')*[B6&4[F27$$\":$H-^J&GzQa/b9*F2$\":!o32ZYS3EK f(p&F27$$\":es3)=fjEV-2m%)F2$\":u5'z_GUzzf%*GlF27$$\":D7O4y(e1b3%3p(F2 $\":g;Uc>k!p!)[M'G(F27$$\":'QVDb2j)3<%4IoF2$\":3Qf'*RV'p()ocbzF27$$\": g_&**QwQNnRU()eF2$\":q%*H4h@>nOO#H&)F27$$\":06.))Hbtv9Xv%\\F2$\":j)pk$ GS5\"R/+m*)F27$$\":'>O#3[K!Q[cD_RF2$\":66\"ySMNIVa<-$*F27$$\":jUG8'o.5 HA)H(GF2$\":UC*[#[N%4:A'\\`*F27$$\":WL(*f)\\\")>i]qbq:F!G;D2'eF2$\" :e2[Z[Lj5gd2/'F27$$!:&ovu![W%*))=#GHlF2$\":nUer&[zsje+')\\F27$$!:asg3- k-3n[\")4(F2$\":u&)pdE\\%*)fr(3m$F27$$!:`WJ/x*Hi9>$QV(F2$\":;u]7i70NTe k@#F27$$!:@2$y#=%fVMZ#p\\(F2$\":0:6mXQbkdea&oFap7$$!:!3ky(4*RI:;C]sF2$ 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It took more than two turns around, in phi, \+ in order to get back to its aphelion. In order to see better how that works; let's make some other plots." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Eq9(100);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&/-%$tau G6#\"$+\"F(/--%$phiG6#F&F'$\":H9E$)fwNa,'>MZ!#C/--%\"rGF-F'$\":9_'*\\7 lG2^P%*='F0/--%%diffG6$F3F&F'$!:#z**>c\\_*H@.Q%[!#E" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot([rhs(Eq9(tau)[1]),rhs(Eq9(tau)[3]),t au=0..300]);" }}{PARA 13 "" 1 "" {GLPLOT2D 494 494 494 {PLOTDATA 2 "6% -%'CURVESG6$7hn7$$\"\"!F)$\"#6F)7$$\":+++++++]7t&pK!#C$\":1RoQY%3%[#RI *4\"!#B7$$\":++++++++DY\"RlF/$\":MkTxXs`%>5v2\"F27$$\":+++++++D'3&o]#F2$\":gfE* ewwnDJVf5F27$$\":++++++](oX*y9$F2$\":%)yX#3(ecB>'QO5F27$$\":++++++]P9C Au$F2$\":qq%p(y]wJv'p55F27$$\":++++++]P*zhdVF2$\":ux\\RKO#)ep)p*z*F/7$ $\":++++++]P>fS*\\F2$\":u_lBXL]tR%\\T%*F/7$$\":++++++](=$f%GcF2$\":%3! pMRm[Tu4([!*F/7$$\":+++++++]#y,\"G'F2$\":8D]!pYicaF$[h)F/7$$\":+++++++ Dr\"zboF2$\":nE5Tc_@p6Q`@)F/7$$\":+++++++](4&G](F2$\":n+>UnUb3LDsv(F/7 $$\":+++++++]7nD:)F2$\":%zYx`[#4c$oa-tF/7$$\":+++++++]-*oy()F2$\":sPmm 2(oo9L@&)oF/7$$\":++++++]PpnsM*F2$\":va!eaJd=OyUPlF/7$$\":+++++++DFOB+ \"!#A$\":W)Q!yYXW%oW;yhF/7$$\":++++++++R5'f5F`q$\":%=qgx)=8)fJ/IfF/7$$ \":++++++vV!QBE6F`q$\":o^zM*e6&f*4d8dF/7$$\":+++++](o4.sb6F`q$\"::9lWD m#oL;KVcF/7$$\":+++++++]\"o?&=\"F`q$\":FyO$GSJ8oQx)e&F/7$$\":+++++](o% =ev@\"F`q$\":qloQHCC(o)foa&F/7$$\":++++++vVb4*\\7F`q$\":#e**f5t&=jpeN_ &F/7$$\":++++++](3dr!G\"F`q$\":+^1X:iCSTc&=bF/7$$\":++++++DJ'=_68F`q$ \":`//;vmWs2%HIbF/7$$\":+++++]PM&[mV8F`q$\":!Q5(zCywm3\\/c&F/7$$\":+++ +++]P%y!eP\"F`q$\":6n+S2LT&Gc.4cF/7$$\":+++++]7G9D`S\"F`q$\":2?f[9G`P \\A+n&F/7$$\":++++++v=WU[V\"F`q$\":c(yXwF$GD$QwYdF/7$$\":++++++]7B>&) \\\"F`q$\":$))oZ$**Htp\"))olfF/7$$\":++++++v$>:mk:F`q$\":&>]G&=h3B&o+n iF/7$$\":++++++DcdQAi\"F`q$\":./;)QS\\sVFm$e'F/7$$\":++++++]PPBWo\"F`q $\":[Sstx$*\\v9?5(pF/7$$\":+++++++]Nm'[RrH(zSO0uF/7$$\":+ +++++](yb^6=F`q$\":)e+P`YF`q$\":40c)*F/7$$\":+++++++vZf\")=#F`q$\":ys>^; %\\F\\W\"o,\"F27$$\":++++++vV?S&[AF`q$\":S.^$**4$H;a^>/\"F27$$\":+++++ +v=Yb;J#F`q$\":?,m&4F./S&*ej5F27$$\":+++++++D;iLP#F`q$\":H5l[+?m3y')*z 5F27$$\":++++++v$fL'zV#F`q$\":mW:`UZ24A!)>4\"F27$$\":+++++](ouE2pCF`q$ \":UNz0x*fjl2%e4\"F27$$\":++++++++*>=+DF`q$\":%4Z$eh;.OhZ%)4\"F27$$\": ++++++D\"ed*>`#F`q$\":9!)yw$RYtNE\")*4\"F27$$\":++++++]i_4Qc#F`q$\":=2 f/1)Gf)*)f)*4\"F27$$\":+++++]7.')f`f#F`q$\":T&>o()o3(\\C0')4\"F27$$\": ++++++vV>5pi#F`q$\":!*pmDE&*od[cg4\"F27$$\":+++++](ou;!fl#F`q$\":RMCIm 'eP')pd#4\"F27$$\":+++++++]:$*[o#F`q$\":TMATd<;P,8!)3\"F27$$\":++++++] 7<[8v#F`q$\":)[BeopkGex\\t5F27$$\":++++++++L'y5GF`q$\":\")p\"Q4?zI@V\" e0\"F27$$\":++++++vV!)fT(GF`q$\":IF7m=]\"*e>;A.\"F27$$\":++++++DcI;[$H F`q$\":!GSt(o+T4$=C05F27$$\"$+$F)$\":T-\"\\w/2,G'exr*F/-%'COLOURG6&%$R GBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q!6\"Fj^l-%%VIEWG6$%(DEFAULTGF__l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 106 "This shows the radial oscillation s well, as a function of tau. We see it started at 11, went down to i ts " }}{PARA 0 "" 0 "" {TEXT -1 94 " minimum of about 5.5, at tau abou t 128 or so, and then back up to 11, at tau about 260 or so." } {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "plot([r hs(Eq9(tau)[1]),rhs(Eq9(tau)[3]),tau=254.8..255.2],numpoints=200);" }} {PARA 13 "" 1 "" {GLPLOT2D 494 494 494 {PLOTDATA 2 "6%-%'CURVESG6$7dw7 $$\"%[D!\"\"$\":Sr^;wf08=,+5\"!#B7$$\":jD^-05?/.@![D!#A$\":-!*)*yl9'f4 7++6F-7$$\":8E_/4=w)G$R![DF1$\":6;6(oNHuL7++6F-7$$\":(Qxa4>QS2*f![DF1$ \":>&4&Gs^x.E,+5\"F-7$$\":g?T#['HRBi!3[DF1$\":`X$*4#*eJmG,+5\"F-7$$\": )oPv],.#)Q75[DF1$\":Et&)Qt&f?78++6F-7$$\":/3;KkGPHN?\"[DF1$\":JtflC6Aa L,+5\"F-7$$\":KkGd9H)\\W,9[DF1$\":=1hvEkf*e8++6F-7$$\":sV([(\\**eHhg\" [DF1$\":Tp5VG`lFQ,+5\"F-7$$\":yc8Fa3xd,\"=[DF1$\":Z>$*G#>?&fS,+5\"F-7$ $\":tX\"He;Le-??[DF1$\":=^3RT[O#H9++6F-7$$\":V&3xHT#[DF1$\":]@XGz0\"Hr9++6F-7$$\":!)f>Ryc8J >i#[DF1$\":#[*[0_tMG\\,+5\"F-7$$\":/3;KkG(pHBG[DF1$\":-dd)4\")z08:++6F 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FcjnFbjn-%+AXESLABELSG6$Q!6\"Fgjn-%%VIEWG6$%(DEFAULTGF\\[o" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "We can see that numerical inaccuracy is \+ setting in, since the orbit should not get larger than 11; nonetheless , the " }}{PARA 0 "" 0 "" {TEXT -1 37 " maximum seems to be at approx. 254.9" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Eq9(254.9);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&/-%$tauG6#$\"%\\D !\"\"F(/--%$phiG6#F&F'$\":*QAVNs:3J.QV:!#B/--%\"rGF/F'$\":Df4%[8k)*))= ++6F2/--%%diffG6$F5F&F'$\"8UJa.aN-[l8m&!#G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 " We see that at this FIRST return to the aphelion, the \+ angle is already up to 15.4, i.e., almost 5 times Pi." }}{PARA 0 "" 0 "" {TEXT -1 96 " Note that this is entirely consistent with our gra ph above, where that many turns were made." }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 1 " " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 " Now I would like to generate \+ an animated plot, so that we can see it \"happening\":" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "for n from 1 to 20 \+ do PP||n:=polarplot([rhs(Eq9(tau)[3]),rhs(Eq9(tau)[2]),tau=0..50*n], s caling=constrained); end do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "display([seq(PP||n,n=1..20)],insequence=true);" }}{PARA 13 "" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6#-%(ANIMATEG667%-%'CURVESG6$7S7 $$\"#6\"\"!$F.F.7$$\":n\"4QYS8?jTN*4\"!#B$\":*[))R%Q7&=:r*e`$!#D7$$\": g@j1#4)[imTx4\"F3$\":KSAy!HL'*4,B4mF67$$\":D_;'fyFJY9w%4\"F3$\":2ANhni +0;Ne+\"!#C7$$\":f#y\"QTU)zpe^!4\"F3$\":vh&*pXfV+3))=N\"FA7$$\"::h35P2 F'fC0&3\"F3$\":ng&[]u$*3)*>w%p\"FA7$$\":GDkV\"*Hkf*p))y5F3$\":,Ca0y+sI ?S4,#FA7$$\":#*4k)p6k&zN\"Rr5F3$\":Tdp[57E4N[iL#FA7$$\":gl!e1hFhbXXi5F 3$\":*\\qHHtV^LX4qEFA7$$\":\"H15hbM@VLN_5F3$\":R$G$)*fN>O;s)**HFA7$$\" :O:fznI%)RNF2/\"F3$\":VfZT8@#fp@cNLFA7$$\":N3VUEi**Qwb%H5F3$\":,cc=;(H j$eYzi$FA7$$\":;bf+&H^['*)=c,\"F3$\":zXs'=;zAc8/`RFA7$$\":yjG02nfcs60+ \"F3$\":*4!zvFk-roYZF%FA7$$\":?Q$>%=&fzC#H\"[)*FA$\"::<;&pN5HFE!*zXFA7 $$\":FqQKx)[E+Y'fp*FA$\":(HN+[L?yzzb_[FA7$$\":C*)\\%*Gd9R%GD.&*FA$\":% QJ$H$3>\"3?_3<&FA7$$\":8Ea&Ga3!o/0,L*FA$\":VED?%*HSiDF^V&FA7$$\":IpIFG n`.%*=u6*FA$\":4'\\*o99.a,]ft&FA7$$\":IV]oc*e'p,u\">*)FA$\":5^*)G.;sHo mf*fFA7$$\":s6Z+$ [A<(FA$\":n?(fRAtF$)>73xFA7$$\":\")pX8Jn58S!G')oFA$\":/^5:ah&yCq[;zFA7 $$\":o7(*GcLhar\"z\"e'FA$\":V'egCe3jW)G87)FA7$$\":t#))4Qg_j70GviFA$\": ))=yLkl+u&G36$)FA7$$\":[=f&oF]T\")f'3(fFA$\":X[aE(HH<$=*G%[)FA7$$\":ie 4!\\X'HU?gTi&FA$\":RH2\"fQYz?$*>k')FA7$$\":,xa&ehlP#QL^I&FA$\":[xJ.qPA lfPV\"))FA7$$\":H;u[%=c%)='Gq&\\FA$\":ic$zm+-42y>i*)FA7$$\":oT29)Hd!Q# pCNYFA$\":K3Q?*QTlN_t%3*FA7$$\":(HV?UL>cwP#pF%FA$\":![LHtt,Ysj'e?*FA7$ $\":MHx`P.vfg2R$RFA$\":ta)Q]`/\\T^C2$*FA7$$\":0p(y$Qff*)o!ppNFA$\":oTt T#4$*>_sy*R*FA7$$\":8I;lJbx;XA%3KFA$\":e0ZEoIi`2NmZ*FA7$$\":v!)))es*\\ N%y+_#GFA$\":jo0*)3e8jYjAa*FA7$$\":[J&)z&z[Iqav^CFA$\":[(*)*4>[)z>/x!f *FA7$$\":$=qyc,g([o`f1#FA$\":.%y[xPGj[#*3D'*FA7$$\":7!op*o7xQQ%**z;FA$ \":))p].kk*FA7$$\": \"z4J5f1nx?98\"*F6$\":<9a9au(GmO(Gj*FA7$$\":\")yE4)*oS(QQ4n9F6$\":BL[#[^#=^zd#f&*FA7$$!:Gd:c0Kf() eBfJ#F6$\":6B$*o0r#y$\\e!*\\*FA7$$!:`4N0a]%pwh\"oP'F6$\":2\\4A65bX%\\S ;%*FA-%'COLOURG6&%$RGBG$\"#5!\"\"F/F/-%(SCALINGG6#%,CONSTRAINEDG-%%VIE WG6$%(DEFAULTGF[\\l7%-F(6$7SF+7$$\":xR'4@fJz%4`Bq#HtWTw1 (F67$$\":VT;l$fUa'3s44\"F3$\":8%Ry[Y\"*eR)G\">8FA7$$\":jD'[\"RfPe-v!z5 F3$\":;<*op]qD1T1-?FA7$$\":%*4z(*\\.%)[jf@1\"F3$\":9`bKtRy;6_.o#FA7$$ \":P7BMc/>ou\\//\"F3$\":jXFZ')QY\"Qi6VLFA7$$\":$f9&zxLae8Hg,\"F3$\":\\ p(Qr6[=A%eQ%RFA7$$\":l\"G)fP:*o\"QXX')*FA$\":l?NOj_reo0#\\XFA7$$\":Md$ fY6**pmCh8&*FA$\"::j&36!><\"zbUa^FA7$$\":%Q()yOwnS#\\(>>\"*FA$\":?OueR \"enQF`LdFA7$$\":&Q9LB9?)[yn#o')FA$\":*3@QDn,HF)G2I'FA7$$\":T%H,Z\\kz' =dUB)FA$\":n.Ncs=Q(pAotnFA7$$\":$*y![ZYKt7@(eq(FA$\":FMb*Q>\"p*=)*>tsF A7$$\":wj0J/%eP5;qMrFA$\":uC#pFq^F*H2kt(FA7$$\":`0%f_*)ec8bsZlFA$\":.& R'G![g.))=@V\")FA7$$\": v!*FA7$$\":Bw?E\"ypl\"e39\"RFA$\":mXqpN2nK!zS8$*FA7$$\":v<,$p*zSo(\\zD KFA$\":*y:ck)oBy!\\Ft%*FA7$$\":9b&Hg+6ms\\%RX#FA$\":M]s`7%GZ=+`!f*FA7$ $\":Qt\\XPC'HB%HPq\"FA$\":qw#H7cw;Ql#Gk*FA7$$\":Fk])pCZA%>Ml4*F6$\":0v A[+sUJL#yK'*FA7$$\":0A.)yum>\"Qviu\"F6$\":#o)yWOo)eEy1j&*FA7$$!:9b`w`R qb$*GD>'F6$\":j!pz[Aa*F A7$$!:@S'=!H-!)o`<@9#FA$\":`8(3s7;OHm]g^3(FA7$$ !:uU(*pYp&)*30H\"f&FA$\":hyJZ@r3(3uc&Q'FA7$$!:tUX#))pz#zt])3hFA$\":?X< >*zdI&*R)yo&FA7$$!:bNB=ZFou'*\\xe'FA$\":Q\"=O'yZjKG0[([FA7$$!:SayL'*)) oKx)QYpFA$\":gq=pMK*)y3D?3%FA7$$!:U_RIAwjb/@dC(FA$\":GuUj:/e0(GQhJFA7$ $!:*)R*4`#e$H_U%)GuFA$\":RU$Rt\"eVndD?D#FA7$$!:uLW#Gz8zb-y2vFA$\":L;V@ v8fdVsxE\"FA7$$!:1Y-Av_0n=]VY(FA$\":BSYb'*QrN`/>&GF67$$!:P#*4.z=(eIN6# G(FA$!:f!R)*3C*=>yn=[(F67$$!:Rc$=$e\")H![0ZnpFA$!:bexYS8D>_W\"H7bk=! 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$!:$z7\\V!=/POBig%FA$!:%=H%o&>]A`*zvK*FA7$$!:XV*G>8tI4xFdHFA$!:]$\\c5+ %*)))\\%Q<5F37$$!:Gz-\"=*G=Z(pa47FA$!:qSW([uM#*G_Co5F3F][lFd[lFh[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "A:='A'; B:='B';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AGF$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"BGF$" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Now, let's go back and \+ look at the one for Mercury:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "val :=[op(Mercb)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$valG7$/%\"AG$\":' 4];)=`uC()*******!#D/%\"BG$\":y`SCrr.$\\Jq]!*!#@" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "B:=rhs(val[2]); A:=rhs(val[1]);m:=1.477;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG$\":y`SCrr.$\\Jq]!*!#@" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG$\":'4];)=`uC()*******!#D" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG$\"%x9!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Leq,r2eq;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6$/-%%diffG 6$-%$phiG6#%$tauGF*,$*&$\":y`SCrr.$\\Jq]!*!#@\"\"\"-%\"rGF)!\"#F0,(-F% 6$F1-%\"$G6$F*\"\"#F0*($\":?9'pKA;o\\F_\">)!# " 0 "" {MPLTEXT 1 0 39 "iniMe rc:=r(0)=rpm, D(r)(0)=0, phi(0)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%(iniMercG6%/-%\"rG6#\"\"!\")+++Y/--%\"DG6#F(F)F*/-%$phiGF)F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "EqMerc:=dsolve(\{Leq,r2eq,in iMerc\},\{r(tau),phi(tau)\},numeric,output=listprocedure);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'EqMercG7&/%$tauGf*6#F'6\"6#%aoCopyright~( c)~1993~by~the~University~of~Waterloo.~All~rights~reserved.GF*9$F*F*F* /-%$phiGF)f*F)6&%$resG%)solnprocG%)outpointG%&ndsolG6#%inCopyright~(c) ~2000~by~Waterloo~Maple~Inc.~All~rights~reserved.GF*C(>%8_EnvDSNumeric SaveDigitsG%'DigitsG>F<\"#D@%/%-_EnvInFsolveG%%trueG>8&-&%&evalfG6#F;6 #F->FD-FGFI>8%f*6#%#mxG6+F5%#dtG%$datG%(odeprocG%#x0G%$valG%$digG%\"nG %\"iG6#%inCopyright~(c)~2002~by~Waterloo~Maple~Inc.~All~rights~reserve d.GE\\s#Q0soln_proceduresF*=F*6#;\"\"\"\"\"%E\\[l%F[o\")KO#[\"\"\"#\") )e!y9\"\"$\"(Kp5$F\\o\")[>f7Q(complexF*%&falseGC/>8$F->FM=F*6#;F[oFaoE \\[l\"F[o=F*6#;F[o\"#9E\\[l/F[o6%/%)datatypeG%'sfloatG/%&orderG%(C_ord erG/%(storageG%,rectangularGF_oFcpFao7$\"\"!F^qF\\oX*%)anythingG6\"F*[ gl%!%!!!\"8\"8!$!!!>[^`r!\"!!\"!!!%!#:!\"!!\"\"&X*FfpFaqF*[gl!!%\"!!\" (\"(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 " We can now use this to c reate a plot of the orbit:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "polarplot([rhs(EqMerc(tau)[3]),rhs(EqMerc(tau)[2 ]),tau=0..10^(12)],scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 495 495 495 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\")+++Y\"\"!$F*F *7$$\":2@*G!oqs8+SK(*H$>f*QNG%!#=7$$\":[/OS)4(z$y]?UXF/$ \":GCrDUl6r%om')zF27$$\":Zv]`][')47F/7$$\":& R*[(4S[u%QC(fVF/$\"::'zrA\"z%fT*4bh\"F/7$$\":*)*=^4p9E+z:CUF/$\":3v#)= j!p0S=Q3?F/7$$\":YR6zip$*pn#etSF/$\":y'G[BZ)e6$\\'3O#F/7$$\":F>AeI3U\\ feQ*QF/$\":tx&[k&G-xi>?r#F/7$$\":BvM=O\"\\CS[>%o$F/$\":MeO')Rp>C%=feIF 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We want to know what is phi when it clo ses back, i.e., " }}{PARA 0 "" 0 "" {TEXT -1 31 " when r gets back to \+ 46 MegaKm." }}{PARA 0 "" 0 "" {TEXT -1 30 " Let's do some radial plo ts:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "p lot([rhs(EqMerc(tau)[1]),rhs(EqMerc(tau)[3]),tau=2.27*10^(12)..2.29*10 ^(12)]);" }}{PARA 13 "" 1 "" {GLPLOT2D 494 494 494 {PLOTDATA 2 "6%-%'C URVESG6$7S7$$\"0++++++F#!\"#$\":f%[E?G30&\\#\\+Y!#<7$$\":LLLLLL$3VfVqA !#7$\":lL*4M;jvN;W+YF-7$$\":nmmmm;H[D:3F#F1$\":/d0/oWe5g*R+YF-7$$\":LL LLLLe0$=CrAF1$\":Zn'[0xvA%za.g%F-7$$\":LLLLLL3RBr;F#F1$\":)Rd0l7syEBJ+ YF-7$$\":nmmmm;zjf)4sAF1$\":p')4R**omwos-g%F-7$$\":LLLLL$e4;[\\sAF1$\" :;&H:=(pf!y#Q-g%F-7$$\":+++++]i'y]!HF#F1$\":V^cBm*)HE-0-g%F-7$$\":LLLL L$ezs$HLF#F1$\":L#o(\\p/P.+vAF1$\":5UIo12pcxs+g%F-7$$\":++++++]Z/N aF#F1$\":Dvx$4,G*3K`+g%F-7$$\":++++++]$fC&eF#F1$\":/Jet>Oq07P+g%F-7$$ \":LLLLL$ez6:BwAF1$\":&p/+`jj$edC+g%F-7$$\":nmmmmm;=C#owAF1$\":=ly/_./ KM7+g%F-7$$\":nmmmmmm#pS1xAF1$\":N@Zzb(H9iU++YF-7$$\":+++++]i`A3vF#F1$ \":1P.)G[V1&\\(***f%F-7$$\":nmmmmmm(y8!zF#F1$\":\"GE$G\\F*znQ***f%F-7$ $\":+++++]i.tK$yAF1$\":7RjgHS/3W#***f%F-7$$\":+++++](3zMuyAF1$\":h$*>& y>%>Yc$***f%F-7$$\":nmmmmm\"H_?t***f%F-7$$\":nmmmm;z ihl&zAF1$\":t*G^Tk%o\"*3.+g%F-7$$\":LLLLLL3#G,**zAF1$\":!3ze&*R9k0=,+Y F-7$$\":LLLLL$ezw5V!G#F1$\":)4-6x)Qq')fB+g%F-7$$\":+++++]PQ#\\\"3G#F1$ \":$)*4oX#)4$o8O+g%F-7$$\":LLLLLLe\"*[H7G#F1$\":!QGr.*pyR0_+g%F-7$$\": +++++++dxd;G#F1$\":!fLlH$>2()3r+g%F-7$$\":++++++D0xw?G#F1$\":Pg$R^`HFe A4+YF-7$$\":+++++]i&p@[#G#F1$\":l7`#*\\is$R^6+YF-7$$\":++++++vgHKHG#F1 $\":aU#3DU5Z3L9+YF-7$$\":nmmmmmmZvOLG#F1$\":x?H&4I\"))**4r,g%F-7$$\":+ +++++]2goPG#F1$\":\\YEGU\"R([O.-g%F-7$$\":LLLLL$eR<*fTG#F1$\":B@%e]_H$ H\"\\B+YF-7$$\":++++++])Hxe%G#F1$\":cQ7+(eDh<>F+YF-7$$\":nmmmm;H!o-*\\ G#F1$\":lpT/F<#=M\"4.g%F-7$$\":+++++]7k.6aG#F1$\":C=&f2ENX@0N+YF-7$$\" :nmmmmm;WTAeG#F1$\":lU\\r#48xSMR+YF-7$$\":+++++]i!*3`iG#F1$\":%=nvsTDi u4W+YF-7$$\":LLLLLLL*zym'G#F1$\":0Go@#[\")pq#*[+YF-7$$\":LLLLLL3N1#4(G #F1$\":#)*QVA#zSF@T0g%F-7$$\":nmmmm;HYt7vG#F1$\":muuqm*oRt_f+YF-7$$\": +++++++xG**yG#F1$\":'>/+lYV-(=Z1g%F-7$$\":nmmmmmT6KU$)G#F1$\":Z-Ye:bB6 K42g%F-7$$\":LLLLLLLbdQ()G#F1$\":$oHX(eK9*ysw+YF-7$$\":+++++]i`1h\"*G# F1$\":kJ>`&GAq`:$3g%F-7$$\":+++++]P?Wl&*G#F1$\":.vX!3Stbpa*3g%F-7$$\"0 ++++++H#F*$\":a0aHPlN(on'4g%F--%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fa[lF` [l-%+AXESLABELSG6$Q!6\"Fe[l-%%VIEWG6$%(DEFAULTGFj[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "plot([rhs(EqMerc(tau)[1]),rhs(EqMerc(tau)[3]) ,tau=2.277*10^(12)..2.28*10^(12)]);" }}{PARA 13 "" 1 "" {GLPLOT2D 494 494 494 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$\"1++++++xA!\"$$\":Z*o)>#Qo*>Z 0+g%!#<7$$\":+++++]i9RlqF#!#7$\":wX&op0ezPU++YF-7$$\":++++]PC#)GArF#F1 $\":#)o[]4`iQ@.+g%F-7$$\":+++++v$eui=xAF1$\":?6$*))\\Jey6-+g%F-7$$\":+ ++++D'3&o]sF#F1$\":HfZ)>>m%R2,+g%F-7$$\":++++](oX*y9tF#F1$\":].#Gd;j7% 4++g%F-7$$\":++++]P9CAutF#F1$\":\"f#\\P\"\\_SQ#****f%F-7$$\":++++]P*zh dVxAF1$\":5d#o')=$)y0%)***f%F-7$$\":++++]P>fS*\\xAF1$\":0rs#))3'y=g(** *f%F-7$$\":++++](=$f%GcxAF1$\":\"=!**H()48%eo***f%F-7$$\":+++++]#y,\"G wF#F1$\":G[$HMxZ%R:'***f%F-7$$\":+++++Dr\"zboxAF1$\":ja6ZdE\"3%e&***f% F-7$$\":+++++](4&G]xF#F1$\":/jVCP*[H**\\***f%F-7$$\":+++++]7nD:yF#F1$ \":EMDEp=!esW***f%F-7$$\":+++++]-*oy(yF#F1$\":-\">$G&G*GB-%***f%F-7$$ \":++++]PpnsMzF#F1$\":]!eCkgI@iO***f%F-7$$\":+++++]siL-!yAF1$\":\"*4k< AY+WH$***f%F-7$$\":++++++!R5'f!yAF1$\":ZBgr_$G:MI***f%F-7$$\":++++]P/Q BE\"yAF1$\":YKnMwmp1z#***f%F-7$$\":++++++:o?&=yAF1$\":@^_q5K3$GE***f%F -7$$\":++++]Pa&4*\\#yAF1$\":$>bX$p'zg2D***f%F-7$$\":++++]7j=_6$yAF1$\" :a!f8dFB`[C***f%F-7$$\":+++++vVy!ePyAF1$\":`-DPk]%)\\W#***f%F-7$$\":++ ++](=WU[VyAF1$\":5'oUsP.)R\\#***f%F-7$$\":+++++DJ#>&)\\yAF1$\":>7\"=]! 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" }}{PARA 0 "" 0 "" {TEXT -1 41 "We can also look at it in the polar mode." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "polarplot([rhs(EqMerc(tau)[3 ]),rhs(EqMerc(tau)[2]),tau=2.27*10^(12)..2.29*10^(12)]);" }}{PARA 13 " " 1 "" {GLPLOT2D 494 494 494 {PLOTDATA 2 "6$-%'CURVESG6$7S7$$\":DDcvk8 deIhvf%!#<$!:LWVqudm\\p()>k\"!#=7$$\":5'y9'\\D0%)f3yf%F*$!:5>-g/rT4Hci b\"F-7$$\":KHB/M&>o\")H,)f%F*$!:D\")=k$*RxE=e;[\"F-7$$\":kNq.8:+?%3B)f %F*$!:XUH$G*o/*R4w(R\"F-7$$\":7*f*>Ls)Rr)*!#>7$$\":$zs%QrP7Mxb#*f%F*$!:u`CB#\\)eT>(RR!*FV7 $$\":$=\\08//vu')Q*f%F*$!:r)eadTSbsdb$=)FV7$$\":Z3\"*))o*p(Qz%\\*f%F*$ !:eCx>cVlg*f%F*$!:%3O$[3>s2()R5e'FV7$$\":(Hu fX0Y7tlp*f%F*$!:EjHPQY8!>d')GdFV7$$\":U(=^`YtOC`x*f%F*$!:H=W%pN*)G3Ei2 \\FV7$$\":^P9#[A#[=IO)*f%F*$!:CqGq(z6jRR%=;%FV7$$\":O'Qqxj!oqv&*)*f%F* $!:=\"QrIs/))o$H]F$FV7$$\":\"y;+)[IS(G]$**f%F*$!:P)4Ypq*=-3KFLEj()!#?7$$\":)=9UyvV^KC***f%F*$!:4s3*f8+/?%o;w#!#@7$$\":'zM\"pEm P@%p)**f%F*$\":n)GS5eB\\5fb0yFhq7$$\":)[s&RV.,!f'o**f%F*$\":;Ll]wZ&*3G 'zB;FV7$$\":,18x3M`ldS**f%F*$\":%H`jiNW:2/:)R#FV7$$\":SJtPr)3Kl\")*)*f %F*$\":a$4bscq@E%*QLKFV7$$\":&o8jQ6g9$zS)*f%F*$\":3Dy*>(GSkYe45%FV7$$ \":8DSRU]79!)z(*f%F*$\":C@gT%oPUg*ph&[FV7$$\":9-nr.%QNxBq*f%F*$\":.AKz4'*f%F*$\":uMhKeD_\\WBW^'FV7$$\":^E.tmD]\" Ho]*f%F*$\":(*)49&HiM'p=uQtFV7$$\":')G.cHq$yVbR*f%F*$\":=T:FGk,8d$HO\" )FV7$$\":%osxQ*Q**Gae#*f%F*$\":B#>qo^^L%p9=-*FV7$$\":WD%f=(R*=\"QB\"*f %F*$\":fN7K05^//ru\")*FV7$$\":d^ZO/N`jYm*)f%F*$\":e.fCS'p+(o(pm5F-7$$ \":taK6Pes*RI\"))f%F*$\":dR*z=\\O)G#HnV6F-7$$\":%*e`ZB())**yIj)f%F*$\" :S\"f_i$)Gj?T#yA\"F-7$$\":D?sPa+?(*3_%)f%F*$\":c=!G,/p:62+28F-7$$\":0m )y'G$f*)>3D)f%F*$\":*[z'3g]'f%F*$\":E'>8pCFq\\L=m>F-7$$\": 4c)f+cKQsTA'f%F*$\":yL&f#3Ug[)o3W?F-7$$\":!Q%3CTci'>;\"ff%F*$\":5\"pu0 Y>j9H9F@F-7$$\":fY%**[gs/93g&f%F*$\":EhEY8FIDsKm?#F-7$$\":LI:XS$*fr5a_ f%F*$\":!3n<*ec(*)\\#\\?H#F--%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fc[lFb[l -%%VIEWG6$%(DEFAULTGFg[l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "There we see the perihelion point---notice that I took out the constraint o n the scaling, so that now the scaling is " }}{PARA 0 "" 0 "" {TEXT -1 88 "very different in the two directions! 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" }}{PARA 0 "" 0 "" {TEXT -1 63 "Therefore I will decide that anywhere in there is cl ose enough." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "EqMerc(2.27 82*10^(12)); EqMerc(2.27827*10^(12)); EqMerc(2.27829*10^(12));EqMerc(2 .27834*10^(12)); EqMerc(2.278345*10^(12)); EqMerc(2.2783455*10^(12));E qMerc(2.278346*10^(12)); " }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&/-%$tau G6#$\"2++++++#yA!\"%F(/--%$phiG6#F&F'$\":KBRqE1.=adDG'!#C/--%\"rGF/F'$ \":V'3\"3E!HT&f#***f%!#\\S4P*Hb)*y!#K" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&/-%$tauG6#$\"4++++++X$yA!\"'F(/--%$phiG6#F&F'$\":Q5xu(oL'=vxJG '!#C/--%\"rGF/F'$\":*ot82!*=GRC***f%!#-f#!#K" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7&/-%$tauG6#$\"4++++++Y$yA!\"'F(/--%$phiG6#F&F'$\":BS=Z z6u\"H?=$G'!#C/--%\"rGF/F'$\":Q>0Zc/`#RC***f%!# " 0 "" {MPLTEXT 1 0 45 "de lt:=evalf(6.283179890518743544231355-2*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%deltG$!4JRpKH%3m;a!#C" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "Multiply this by 180/Pi to change to degrees and then by \+ 3600 to change to seconds of arc:" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "deltsec:=delt*3600*180/evalf(Pi);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%(deltsecG$!:_D9Odut#*\\ms6\"!#C" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "So this is a precession (negative) of -1.117 seconds per year." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "I also decide to determine what the value of tau is , for the one orbit, in years, rather than kilometers." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "secyr:=365.25*86400; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&secyrG$\"+++wbJ!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "2.2783455*10^(12)/c/secyr;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#$\":hG(4`Hip$48#3C!#D" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "polarplot([rhs(EqMerc(tau)[3]),rhs( EqMerc(tau)[2]),tau=0..2.25*10^(13)],scaling=constrained);" }}{PARA 13 "" 1 "" {GLPLOT2D 494 494 494 {PLOTDATA 2 "6%-%'CURVESG6$7]fl7$$\") +++Y\"\"!$F*F*7$$\"+z9FnX!\"#$\"+SNm;g!\"$7$$\"+QK$)pWF/$\"+t2#[>\"F/7 $$\"+eq()4VF/$\"+w&)Gr()F2$\"+J(3aH&F/7$$ !+;jPY7F2$\"+AGHqbF/7$$!+uI%y6\"F/$\"+E/vmcF/7$$!+R\\3d>F/$\"+Q\\R&F/7$$!+slh(>%F/$\"+<7 `V[F/7$$!+5r&4#[F/$\"+3![cT%F/7$$!+pP*=P&F/$\"+&R38#RF/7$$!+PW=YeF/$\" +o\\\"4P$F/7$$!+v)*fSiF/$\"+tcRuFF/7$$!+\"ewde'F/$\"+8$Q-1#F/7$$!+C2]C oF/$\"+W'>FJ\"F/7$$!+c(=_&pF/$\"+S:&RW&F27$$!+#4Dr(pF/$!+t`kCBF27$$!+H u3!*oF/$!+_go05F/7$$!+yxj%p'F/$!+!)e0j!=$F/7$$!+*[42Z&F/$!+E_&y\"QF/7$$!+)GY!e[F/$!+.%*>'Q%F /7$$!+.$\\A:%F/$!+UBAq[F/7$$!+LqPhLF/$!+MW8a_F/7$$!+Db='\\#F/$!+UzQ@bF /7$$!+Di$4d\"F/$!+iU*\\l&F/7$$!+f>\"H/'F2$!+H;/QcF/7$$\"+cuq'z$F2$!+$f pZX&F/7$$\"+pZ-Y8F/$!+c.S%4&F/7$$\"+8VwiAF/$!+(*z(*[XF/7$$\"+*=@r3$F/$ !+f&\\&>QF/7$$\"+P**[sPF/$!+.(4%>HF/7$$\"+qg`[SF/$!+UKi8CF/7$$\"+'3!3t UF/$!+-#zr(=F/7$$\"+!>^;W%F/$!+a8=;8F/7$$\"+Rst]XF/$!+&\\LeP(F27$$\"+D =*zf%F/$!+R.v*[\"F27$$\"+C/Q#e%F/$\"+/aj;WF27$$\"+!QWU]%F/$\"+())*GE5F /7$$\"+S`GlVF/$\"+w62(f\"F/7$$\"+_>YdRF/$\"+it`%f#F/7$$\"+$z;!zLF/$\"+ Zus![$F/7$$\"+&pN]m#F/$\"+6^7GUF/7$$\"+8(HI&=F/$\"+G]4@[F/7$$\"+vb@!z* F2$\"++$pUD&F/7$$\"+IOr>v!\"%$\"+_a2IbF/7$$!+;Ez5$)F2$\"+7oCccF/7$$!+o JL<)\\&F/7$$!+$*Hl:MF/$\"+sRNK_F/7$$ !+#*p)z;%F/$\"+*3Q5'[F/7$$!+NEYU[F/$\"+%o[')R%F/7$$!+l30KaF/$\"+')R'*e QF/7$$!+7z[JfF/$\"+Ar4bKF/7$$!+j['oL'F/$\"+\"\\@&*f#F/7$$!+R%e`k'F/$\" +D:8/>F/7$$!+.KUgoF/$\"+Y%[_:\"F/7$$!+2:TopF/$\"+Aq<$)QF27$$!+Y&o'opF/ $!+k#)GYQF27$$!+)z$>hoF/$!+_1j^6F/7$$!+R$Rmk'F/$!+;@i+>F/7$$!+P=PEjF/$ !+4FK>EF/7$$!+4)fD!fF/$!+Zl3&H$F/7$$!+@qOy`F/$!+BAr9RF/7$$!+A_QgZF/$!+ 0qpiWF/7$$!+I[1`SF/$!+qvqE\\F/7$$!+-VRkKF/$!+%zA:H&F/7$$!+R2/0CF/$!+Vf FTbF/7$$!+P%Q!*[\"F/$!+!R)pfcF/7$$!+v_^Y`F2$!+quoIcF/7$$\"+[/iYVF2$!+' Q9$RaF/7$$\"+(4i%*Q\"F/$!+E#eM2&F/7$$\"+/Li=BF/$!+W!Qy]%F/7$$\"+Dq.\\J F/$!+E@GF/7$$\"+,E \">5%F/$!+BYN*H#F/7$$\"+hl%zJ%F/$!+TG/ZBNA'F27$$\"+4IelWF/$\"+U5&Q@\"F/7$$\"+sm//VF/$\"+(>w%)y\"F/7$$\"+ tw%Q3%F/$\"+h#p(QBF/7$$\"+/Bd4QF/$\"+7B8eGF/7$$\"+Uih6KF/$\"+*)=b\"o$F /7$$\"+sJi&\\#F/$\"+!*fVqVF/7$$\"+)f(\\&p\"F/$\"+1U+8\\F/7$$\"+wjnK%)F 2$\"+(GDgI&F/7$$!+:6gkKFd\\l$\"+$>pEb&F/7$$!+.x,\"3*F2$\"+[*Q/m&F/7$$! 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