{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 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 "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "ErVenusM.mws Revised trip from Earth to Venus 10/30/98" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 58 "There was a subtle problem with p as a fu nction of theta1." }}{PARA 0 "" 0 "" {TEXT -1 23 "See the comments bel ow." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "T he constants built into this calculation assume a body moving in the g ravitational field of the Sun" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart;with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "twopi:=2*Pi*(1-eccen^2)^(-3/ 2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "g:=int((1+eccen*cos( x))^(-2),x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "f:=piecewis e(y=Pi,twopi-subs(x=2*Pi-y,g),y>=2*Pi,twopi+subs(x=y -2*Pi,g)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "plot(subs(ecc en=1/2,f),y=0..7,title=`f vs y, e=1/2`):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "deltatheta:=Pi;rInit:=1;rFinal:=0.7233;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%+deltathetaG%#PiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&rInitG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'r FinalG$\"%Ls!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "values: =\{r1=rInit,r2=rFinal,dtheta=deltatheta\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'valuesG<%/%#r1G\"\"\"/%#r2G$\"%Ls!\"%/%'dthetaG%#PiG " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "eq1:=r2/r1 = (1+ecc*cos (theta1))/(1+ecc*cos(theta1+dtheta));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq1G/*&%#r2G\"\"\"%#r1G!\"\"*&,&F(F(*&%$eccGF(-%$cosG6#%'theta1G F(F(F(,&F(F(*&F.F(-F06#,&F2F(%'dthetaGF(F(F(F*" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 23 "eq1a:=subs(values,eq1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eq1aG/$\"%Ls!\"%*&,&\"\"\"F+*&%$eccGF+-%$cosG6#%'the ta1GF+F+F+,&F+F+*&F-F+-F/6#,&F1F+%#PiGF+F+F+!\"\"" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 22 "eccn:=solve(eq1a,ecc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%eccnG,$*$-%$cosG6#%'theta1G!\"\"$!+T.k0;!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "p1 := r1*(1+eccn*cos(theta1) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#p1G,$%#r1G$\"+f'fVR)!#5" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p:=subs(values,p1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG$\"+f'fVR)!#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "h:=p^(3/2)*subs(eccen=eccn,y=theta,f)/(2* Pi):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "h is the time to travel f rom angle zero to angle theta1 with given parameters p and e." }} {PARA 0 "" 0 "" {TEXT -1 90 "Notice that p is a function of theta1. Wh en we subtract the 2 times below, p as a function" }}{PARA 0 "" 0 "" {TEXT -1 64 "of theta1 must not change when the limits of integration \+ change." }}{PARA 0 "" 0 "" {TEXT -1 78 "By subtracting the two times, \+ we get the travel time through angle delta-theta" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Ttravel:=subs(theta=theta1+deltatheta,h)-subs (theta=theta1,h):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "plot(T travel,theta1=-Pi..Pi,time=0..1,title=`time vs initial angle`);" }} {PARA 13 "" 1 "" {INLPLOT "6&-%'CURVESG6$7_o7$$!+aEfTJ!\"*$!+iv8**R!#5 7$$!+$z7t8$F*$\"+l`f&*RF-7$$!+KH.LJF*$\"+T)e?*RF-7$$!+rIvGJF*$\"+Xy_)) RF-7$$!+5KZCJF*$\"+NA+&)RF-7$$!+)[8f6$F*$\"+Nm'z(RF-7$$!+lPN2JF*$\"+24 &4(RF-7$$!+@VB!4$F*$\"+rY(p&RF-7$$!+v[6tIF*$\"+g[1VRF-7$$!+')f()QIF*$ \"+D/T:RF-7$$!+&4PY+$F*$\"+i3#z)QF-7$$!+Y\\0XHF*$\"+BOGSQF-7$$!+'zsa)G F*$\"+V-h#z$F-7$$!+y+Y^FF*$\"+;fZ$o$F-7$$!+P!flh#F*$\"+mWWnNF-7$$!+d\" *H#[#F*$\"+3ehSMF-7$$!+iI#yN#F*$\"+Q6p1LF-7$$!+w_$*GAF*$\"+Nh/UJF-7$$! +U$Rc4#F*$\"+MoFFHF-7$$!+\"*3xi>F*$\"+!y%QOEF-7$$!+d$*4E=F*$\"+/ND&=#F -7$$!1+++&**=dq\"!#:$\"1yjQU,!f^\"!#;7$%%FAILGFdr7$$!+C$))HI\"F*$\"+%= dxI\"F*7$$!+qH5)G\"F*$\"+S=wO@\"F*$\"+cK\"HV)F-7$$!+'[0R =\"F*$\"+cjLbxF-7$$!+uWIU5F*$\"+r\\%[4'F-7$$!*V)\\B#*F*$\"+zCkcaF-7$$! *!49GyF*$\"+[1J:]F-7$$!*08If'F*$\"+B9@_ZF-7$$!*0#)yB&F*$\"+q%>o`%F-7$$ !*(oZZRF*$\"+;W1uVF-7$$!*Wu5g#F*$\"+A^SJUF-7$$!*rcYO\"F*$\"+eA&f6%F-7$ $!(:85$F*$\"+mzq,SF-7$$\"*pnUN\"F*$\"+$pX\"*)QF-7$$\"*ue,c#F*$\"+iOp#z $F-7$$\"*5aD'QF*$\"+Ttp'o$F-7$$\"*'f03_F*$\"+hMArNF-7$$\"*-nV_'F*$\"+5 !puW$F-7$$\"*mlzz(F*$\"+j\")G6LF-7$$\"*!)z?@*F*$\"+n=')HJF-7$$\"+ECF[5 F*$\"+^R)H#HF-7$$\"+V3%R=\"F*$\"+Dx\"Hi#F-7$$\"+$fwoI\"F*$\"+Kdy?AF-Fc r7$$\"+Na;H=F*$\"+xJ?99F*7$$\"+*)z2Y=F*$\"+9yTP7F*7$$\"+V0*H'=F*$\"+Hn )46\"F*7$$\"+(4.*z=F*$\"+<*pi,\"F*7$$\"+^c\"o*=F*$\"+*fPvU*F-7$$\"+f2k I>F*$\"+xYzh$)F-7$$\"+meYk>F*$\"+\"4Npi(F-7$$\"+k5iH?F*$\"+g\"4eq'F-7$ $\"+hix%4#F*$\"+&RGr7'F-7$$\"+hu.GAF*$\"+&>l=U&F-7$$\"+#>&>gBF*$\"+JQx 6]F-7$$\"+?Wj\"[#F*$\"+\\(4Mv%F-7$$\"+\"[<3i#F*$\"+/*GE`%F-7$$\"+pWIXF F*$\"+%QLeP%F-7$$\"+kN.yGF*$\"+i\"4[B%F-7$$\"+VA20IF*$\"+**4+;TF-7$$\" +aEfTJF*$\"+hv8**RF--%'COLOURG6&%$RGBG$\"#5!\"\"\"\"!Fc`l-%&TITLEG6#%6 time~vs~initial~angleG-%+AXESLABELSG6$%'theta1G%%timeG-%%VIEWG6$;F(Fh_ l;Fc`l$\"\"\"Fc`l" 2 285 285 285 2 0 1 0 2 9 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0" 15 } {VIEWOPTS 1 1 0 1 1 1803 }