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{SECT 0 {PARA 0 "" 0 "" {TEXT 209 0 "" }}{PARA 18 "" 0 "" {TEXT 214 
16 "Planetary Motion" }}{PARA 19 "" 0 "" {TEXT 215 49 "Worksheet by Mi
ke May, S.J. \302\254\302\2512006- maymk@slu.edu" }}{PARA 0 "" 0 "" 
{TEXT 209 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}
}{PARA 210 "" 0 "" {TEXT 208 220 "One use of parameterized curves is t
o plot planetary motion.  That is actually a good example to study sin
ce one could make a good case for the claim that Newton invented calcu
lus to solve problems of planetary motion.  " }}{PARA 210 "" 0 "" 
{TEXT 208 52 "This is a demonstration worksheet without exercises." }}
{PARA 0 "" 0 "" {TEXT 209 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 216 32 
"Parameterizing elliptical motion" }{TEXT 216 0 "" }}{PARA 0 "" 0 "" 
{TEXT 209 197 "The first approach to planetary motion in the class is \+
to look at trying to parameterize the elliptical paths of the planets.
  We want to start with parameterizing circular motion and modifying i
t." }}{PARA 0 "" 0 "" {TEXT 209 0 "" }}{PARA 0 "" 0 "" {TEXT 209 95 "I
f we want to move around a circle of radius r and period omega the eas
iest parameterization is" }}{PARA 0 "" 0 "" {TEXT 209 56 "x(t) = a*cos
(2*Pi*t/omega),  y(t) = a*sin(2*Pi*t/omega)." }}{PARA 0 "" 0 "" {TEXT 
209 0 "" }}{PARA 0 "" 0 "" {TEXT 209 65 "For the earth, r=93 in millio
ns of miles, and omega = 1 in years." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "a := 93:  omeg := 1:\n" }{MPLTEXT 1 0 60 "x := t -> a
*cos(2*Pi*t/omeg):  y := t ->a*sin(2*Pi*t/omeg):\n" }{MPLTEXT 1 0 28 "
plot([x(t), y(t), t=0..1] );" }{MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" 
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}{TEXT 217 0 "" }}}{PARA 0 "" 0 "" {TEXT 209 140 "Next we want to take
 into account that the orbit of the earth is flattened into an ellipse
.  The eccentricity of the earth's orbit is .02.  " }}{PARA 0 "" 0 "" 
{TEXT 209 0 "" }}{PARA 0 "" 0 "" {TEXT 209 77 "That means that the sum
 is shifted c = e*a = .02*93 million miles off center." }}{PARA 0 "" 0
 "" {TEXT 209 85 "It also means that the short axis is b = sqrt(a^2-c^
2) = a*sqrt(1-e^2) million miles." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
 1 0 12 "e := .02:  \n" }{MPLTEXT 1 0 10 "c := a*e;\n" }{MPLTEXT 1 0 
19 "b := a*sqrt(1-e^2);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "$\"$'=!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+9)R\")H*!
\")" }}}{PARA 0 "" 0 "" {TEXT 209 59 "This leads to a new parameteriza
tion of the earth's motion." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "a := 93:  omeg := 1:\n" }{MPLTEXT 1 0 46 "e := .02:    c := a*e:  \+
  b := a*sqrt(1-e^2):\n" }{MPLTEXT 1 0 36 "x := t -> a*cos(2*Pi*t/omeg
) + c:  \n" }{MPLTEXT 1 0 30 "y := t -> b*sin(2*Pi*t/omeg):\n" }
{MPLTEXT 1 0 28 "plot([x(t), y(t), t=0..1] );" }{MPLTEXT 1 0 0 "" }}
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45.0 45.0 0 0 "Curve 1" }}{TEXT 217 0 "" }}}{PARA 0 "" 0 "" {TEXT 209 
50 "This still looks like a circle to the unaided eye." }}{PARA 0 "" 0
 "" {TEXT 209 0 "" }}{PARA 0 "" 0 "" {TEXT 209 72 "If we look at the o
rbit of Mercury, a is 36 million miles and e is .21.\n" }{TEXT 209 
177 "We also need to compute the period, which will be (36/93)^(3/2). \+
 (The period is the size of the orbit, measured in AUs (The size of th
e earth's orbit) raised to the 3/2 power.)" }}{EXCHG {PARA 0 "> " 0 ""
 {MPLTEXT 1 0 42 "a := 36:   omeg := evalf((a/93.0)^(3/2));\n" }
{MPLTEXT 1 0 46 "e := .21:    c := a*e:    b := a*sqrt(1-e^2):\n" }
{MPLTEXT 1 0 36 "x := t -> a*cos(2*Pi*t/omeg) + c:  \n" }{MPLTEXT 1 0 
30 "y := t -> b*sin(2*Pi*t/omeg):\n" }{MPLTEXT 1 0 51 "plot([x(t), y(t
), t=0..omeg], scaling=constrained);" }{MPLTEXT 1 0 0 "" }}{PARA 11 ""
 1 "" {XPPMATH 20 "$\"+QRS3C!#5" }}{PARA 13 "" 1 "" {TEXT 217 0 "" }
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$!2V2vqRu(yD!#:$!2U#=2\"3ffK\"!#:7$$!1M233qqmB!#9$!1xMT^:M^<!#97$$!1E[
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#9$!1hZT(>$\\]K!#97$$\"2j4)31#)=dD!#:$!2UpqbC+v/$!#:7$$\"2ED!fUhObH!#:
$!2QXYnW)\\'y#!#:7$$\"/8,Q\"[<J$!#7$!0UCQ[W)yC!#87$$\"2v*fu?U1+O!#:$!2
Pwcmc')y:#!#:7$$\"2M54]]h(yQ!#:$!1T@Hj'[7v\"!#97$$\"2/i-DM+q2%!#:$!2Wy
\\ry-'e8!#:7$$\"1M\"oC*zoJU!#9$!1HU#ynI&p\"*!#:7$$\"1Ykq#*R]AV!#9$!1*=
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\"!!\"\"$\"\"!!\"\"-%*AXESSTYLEG6#%'NORMALG-%(SCALINGG6#%,CONSTRAINEDG
" 1 2 2 1 10 1 2 6 0 4 1 1.0 45.0 45.0 0 0 "Curve 1" }}{TEXT 217 0 "" 
}}}{PARA 0 "" 0 "" {TEXT 209 51 "For comparison we plot a circle at th
e same center." }{TEXT 209 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 37 "xc := t -> a*cos(2*Pi*t/omeg) + c:  \n" }{MPLTEXT 1 0 31 "yc := \+
t -> a*sin(2*Pi*t/omeg):\n" }{MPLTEXT 1 0 80 "plot(\{[xc(t), yc(t), t=
0..omeg], [x(t), y(t), t=0..omeg]\}, scaling=constrained);" }{MPLTEXT 
1 0 0 "" }}{PARA 13 "" 1 "" {TEXT 217 0 "" }{GLPLOT2D 400 400 400 
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31#)=dD!#:$!1dabS^+<J!#97$$\"2ED!fUhObH!#:$!2vM6gi]+&G!#:7$$\"/8,Q\"[<
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!#:$!2vtp:r*=\"z\"!#:7$$\"2/i-DM+q2%!#:$!2t<!=*o)e*Q\"!#:7$$\"1M\"oC*z
oJU!#9$!1pZeR=my$*!#:7$$\"1Ykq#*R]AV!#9$!1LUo>B[**[!#:7$$\"%cV!\"#$\"2
;r*f1>8U6!#C-%&COLORG6&%$RGBG$\"2'\\_'eM%yg>!#<$\"1ij9.e@R!)!#;$\"2'
\\_'eM%yg>!#<-%*AXESSTYLEG6#%'NORMALG-%(SCALINGG6#%,CONSTRAINEDG" 1 2 
2 1 10 1 2 6 0 4 1 1.0 45.0 45.0 0 0 "Curve 1" "Curve 2" }}{TEXT 217 
0 "" }}}{PARA 0 "" 0 "" {TEXT 209 75 "It is still very close to circul
ar in orbit, even if noticeably off center." }}{PARA 0 "" 0 "" {TEXT 
209 0 "" }}{PARA 0 "" 0 "" {TEXT 209 59 "For Halley's comment the numb
ers are that a=1680 and e=.97." }{TEXT 209 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 44 "a := 1680:   omeg := evalf((a/93.0)^(3/2));\n" }
{MPLTEXT 1 0 46 "e := .97:    c := a*e:    b := a*sqrt(1-e^2):\n" }
{MPLTEXT 1 0 36 "x := t -> a*cos(2*Pi*t/omeg) + c:  \n" }{MPLTEXT 1 0 
30 "y := t -> b*sin(2*Pi*t/omeg):\n" }{MPLTEXT 1 0 51 "plot([x(t), y(t
), t=0..omeg], scaling=constrained);" }{MPLTEXT 1 0 0 "" }}{PARA 11 ""
 1 "" {XPPMATH 20 "$\"+Cy%yn(!\")" }}{PARA 13 "" 1 "" {TEXT 217 0 "" }
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\"2%H`W!y;ET\"!#8$!1%f)z\\W&*\\S!#87$$\"1_oZ%fjWi\"!#7$!1-'*R$\\VT3%!#
87$$\"2$H=A0v$f%=!#8$!2D;V$)fe,0%!#97$$\"2ub)oWp%)e?!#8$!2(***Rr!4g[R!
#97$$\"1H,TFT,uA!#7$!2kR\"RscvrP!#97$$\"2ELta\\a,Z#!#8$!2M**R^$*4i`$!#
97$$\"2&QZ)e'3(fl#!#8$!/OL8GNLK!#67$$\"2#zg!QY#GAG!#8$!2&)*RF%>ij(G!#9
7$$\"2EM[\"p*Ho&H!#8$!2_h[S9MR]#!#97$$\"2B@%*=q))o3$!#8$!2X:GA6'3K?!#9
7$$\"1jJ[f,SzJ!#7$!2y1'oERZw:!#97$$\"2C&fdHxe^K!#8$!22%)3)R%**R1\"!#97
$$\"29J+)H&oRH$!#8$!1c&**)*y6%eb!#97$$\"&'4L!\"\"$!2o[vkR_j1\"!#B-%&CO
LORG6&%$RGBG$\"#5!\"\"$\"\"!!\"\"$\"\"!!\"\"-%*AXESSTYLEG6#%'NORMALG-%
(SCALINGG6#%,CONSTRAINEDG" 1 2 2 1 10 1 2 6 0 4 1 1.0 45.0 45.0 0 0 "C
urve 1" }}{TEXT 217 0 "" }}}{PARA 0 "" 0 "" {TEXT 209 143 "This orbit \+
is clearly an ellipse.  It is worth noting that our model would have t
he period of Mercury be 88 days and Haley's comet be 77 years." }{TEXT
 209 0 "" }}{PARA 0 "" 0 "" {TEXT 209 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 216 16 "Relative m
otions" }{TEXT 216 0 "" }}{PARA 210 "" 0 "" {TEXT 209 244 "The next qu
estion of interest is to plot relative motion for two planets.  In par
ametric form, we simply take the difference of the two parameterizatio
ns.  Not that we want to take our time period to be at least the longe
r of the two periods.  " }}{PARA 210 "" 0 "" {TEXT 209 0 "" }}{PARA 
210 "" 0 "" {TEXT 209 40 "Consider Mercury and the earth together." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "am := 36:   omegm := evalf((
am/93.0)^(3/2)):\n" }{MPLTEXT 1 0 53 "em := .21:    cm := am*em:    bm
 := am*sqrt(1-em^2):\n" }{MPLTEXT 1 0 40 "xm := t -> am*cos(2*Pi*t/ome
gm) + cm:  \n" }{MPLTEXT 1 0 33 "ym := t -> bm*sin(2*Pi*t/omegm):\n" }
{MPLTEXT 1 0 45 "ae := 93:   omege := evalf((ae/93.0)^(3/2)):\n" }
{MPLTEXT 1 0 53 "ee := .02:    ce := ae*ee:    be := ae*sqrt(1-ee^2):
\n" }{MPLTEXT 1 0 40 "xe := t -> ae*cos(2*Pi*t/omege) + ce:  \n" }
{MPLTEXT 1 0 33 "ye := t -> be*sin(2*Pi*t/omege):\n" }{MPLTEXT 1 0 31 
"tmax := max(2*omegm, 2*omege);\n" }{MPLTEXT 1 0 69 "plot([xm(t) - xe(
t), ym(t) - ye(t), t=0..tmax], scaling=constrained);" }{MPLTEXT 1 0 0 
"" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+++++?!\"*" }}{PARA 13 "" 1 "" 
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SSTYLEG6#%'NORMALG-%(SCALINGG6#%,CONSTRAINEDG" 1 2 2 1 10 1 2 6 0 4 1 
1.0 45.0 45.0 0 0 "Curve 1" }}{TEXT 217 0 "" }}}{PARA 210 "" 0 "" 
{TEXT 209 118 "Looking from one planet to the other it appears that th
e planet is stopping and starting with some backing up as well." }}
{PARA 210 "" 0 "" {TEXT 209 0 "" }}{PARA 210 "" 0 "" {TEXT 209 61 "Con
sider what happens if we try the earth and Halley's comet." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "am := 1680:   omegm := evalf((am/93
.0)^(3/2)):\n" }{MPLTEXT 1 0 53 "em := .97:    cm := am*em:    bm := a
m*sqrt(1-em^2):\n" }{MPLTEXT 1 0 40 "xm := t -> am*cos(2*Pi*t/omegm) +
 cm:  \n" }{MPLTEXT 1 0 33 "ym := t -> bm*sin(2*Pi*t/omegm):\n" }
{MPLTEXT 1 0 45 "ae := 93:   omege := evalf((ae/93.0)^(3/2)):\n" }
{MPLTEXT 1 0 53 "ee := .02:    ce := ae*ee:    be := ae*sqrt(1-ee^2):
\n" }{MPLTEXT 1 0 40 "xe := t -> ae*cos(2*Pi*t/omege) + ce:  \n" }
{MPLTEXT 1 0 33 "ye := t -> be*sin(2*Pi*t/omege):\n" }{MPLTEXT 1 0 31 
"tmax := max(2*omegm, 2*omege);\n" }{MPLTEXT 1 0 69 "plot([xm(t) - xe(
t), ym(t) - ye(t), t=0..tmax], scaling=constrained);" }{MPLTEXT 1 0 0 
"" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"+l&pb`\"!\"(" }}{PARA 13 "" 1 ""
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2jMe)e^VZK!#8$!11z7'H*=a&)!#97$$\"1$e#[-$zPI$!#7$!2JU*[`=sL5!#97$$\"2'
*)zw+KThL!#8$!17_4*4FHF)!#97$$\"2mdh#)[+mR$!#8$!2XD6T=$zrI!#:7$$\"1C]J
Y![[R$!#7$\"196dPp$pD$!#9-%&COLORG6&%$RGBG$\"#5!\"\"$\"\"!!\"\"$\"\"!!
\"\"-%*AXESSTYLEG6#%'NORMALG-%(SCALINGG6#%,CONSTRAINEDG" 1 2 2 1 10 1 
2 6 0 4 1 1.0 45.0 45.0 0 0 "Curve 1" }}{TEXT 217 0 "" }}}{PARA 0 "" 0
 "" {TEXT 209 158 "The motion is obviously wrong.  We have neglected t
hat fact that planets move at different speeds on their orbits.  They \+
go faster when they are near the sun." }{TEXT 209 0 "" }}{EXCHG {PARA 
202 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 216 
36 "Finding a path determined by gravity" }{TEXT 216 0 "" }}{PARA 0 ""
 0 "" {TEXT 209 220 "If we are to look at planetary motion from calcul
us, what we know is that the only force that acts on the planets is gr
avity and that it is in the direction of the sun with a force inversel
y proportional to the distance." }{TEXT 209 0 "" }}{PARA 0 "" 0 "" 
{TEXT 209 0 "" }}{PARA 210 "" 0 "" {TEXT 209 48 "In terms of different
ial equations this becomes:" }}{PARA 210 "" 0 "" {TEXT 209 28 "x'' = -
x/r^3,  y'' = -y/r^3." }}{PARA 210 "" 0 "" {TEXT 209 65 "(Note that x^
2 + y^2 = r^2 and that sqrt((x'')^2+(y'')^2)=1/r^2.)" }}{PARA 210 "" 0
 "" {TEXT 209 0 "" }}{PARA 210 "" 0 "" {TEXT 209 209 "We have only wor
ked with vector fields that are first order differential equations and
 this system is second order.  We handle that problem by defining u=x'
 and v=y' to get a system of 4 first order equations." }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 44 "x:='x':  y:= 'y': u :='u': v:= 'v': t:= '
t':" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "O
DEa := diff(x(t),t)=u(t):\n" }{MPLTEXT 1 0 27 "ODEb := diff(y(t),t)=v(
t):\n" }{MPLTEXT 1 0 50 "ODEc := diff(u(t),t)=-x(t)/(x(t)^2+y(t)^2)^(3
/2):\n" }{MPLTEXT 1 0 50 "ODEd := diff(v(t),t)=-y(t)/(x(t)^2+y(t)^2)^(
3/2):\n" }{MPLTEXT 1 0 33 "eqns := [ODEa, ODEb, ODEc, ODEd];" }
{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "7&/-I%diffG%*protect
edG6$-I\"xG6\"6#I\"tGF*F,-I\"uGF*F+/-F%6$-I\"yGF*F+F,-I\"vGF*F+/-F%6$F
-F,,$*&F(\"\"\"),&*$)F(\"\"#F;F;*$)F2F@F;F;#\"\"$F@!\"\"FE/-F%6$F4F,,$
*&F2F;F<FEFE" }}}{PARA 210 "" 0 "" {TEXT 208 93 "We then specify initi
al conditions and use the command dsolve to produce a solution functio
n." }{TEXT 208 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "ICx :=
 x(0) = 1: \n" }{MPLTEXT 1 0 17 "ICy := y(0) = 0:\n" }{MPLTEXT 1 0 17 
"ICu := u(0) = 0:\n" }{MPLTEXT 1 0 17 "ICv := v(0) = 1:\n" }{MPLTEXT 
1 0 66 "solcurve := dsolve(\{ODEa, ODEb, ODEc, ODEd, ICx, ICy, ICu, IC
v\}, \n" }{MPLTEXT 1 0 40 "     [x(t), y(t), u(t), v(t)], numeric);" }
{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I(x_rkf45G6\"6)I
$resGF%I%dataGF%I%varsGF%I)solnprocGF%I)outpointGF%I&ndsolGF%I\"iGF%6#
IinCopyright~(c)~2000~by~Waterloo~Maple~Inc.~All~rights~reserved.GF%F%
C*>I8_EnvDSNumericSaveDigitsGF%I'DigitsGF%>F3\"#:@%/I-_EnvInFsolveGF%I
%trueG%*protectedG>F+-&I&evalfGF:6#F2F#>F+-F>F#>F(-I%evalGF:6#&I<dsolv
e/numeric/data/modulesGI(_syslibGF%6#\"\"\">F)-_F(I$GetGF%6#Q(sysvarsF
%>F*-FM6#Q/soln_procedureF%@$4-I%typeGF:6$F+.I(numericGF:C$@,-I'member
GF:6$F$7+Q&startF%.I&startGF%Q'methodF%.I'methodGF%Q%leftF%Q&rightF%Q)
leftdataF%Q*rightdataF%Q+enginedataF%C$>F'-F*6#-I(convertGF:6$F$.I'str
ingGF:@&-FX6$F'.I&arrayGF:O-FD6$F'FJ0F'Q)procnameF%OF'-Fin6$F$7&Q%last
F%.I%lastGF%Q(initialF%.I(initialGF%C$Fho@$-FX6$F'.I%listGF:O7#-I$seqG
F:6$/&F)6#,&F-FJFJFJ&F'Far/F-;\"\"!\"\"%3-FX6$F+I\"=GF:-Fin6$-I$lhsGF:
6#F+7$FaqFbqC$@%-FX6$-I$rhsGF:F`sFhq>F'-F*6#/Faq7$Ffr-I#opGF:6#FfsYQKi
nitial~values~must~be~specified~in~a~listF%Feq/F+Q.solnprocedureF%O-FD
6#F*/F+FPOF)@%0%)procnameGI(unknownGF%O-.F[uF#C$>F,-I(pointtoGF:6#&-FM
6#Q0soln_proceduresF%6#FfrO-.F,F#Z%C$>F'-F*F`sF[rF%YF%F%F%F%" }}}
{PARA 0 "" 0 "" {TEXT 209 74 "Then we can look at where the planet is \+
at times Pi/2, Pi, 3Pi/2, and 2Pi." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "P1 := solcurve(Pi/2);\n" }{MPLTEXT 1 0 20 "P2 := solc
urve(Pi);\n" }{MPLTEXT 1 0 24 "P3 := solcurve(3*Pi/2);\n" }{MPLTEXT 1 
0 21 "P4 := solcurve(2*Pi);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "7'/I\"tG6\"$\"0!\\zEjzq:!#9/-I\"xGF%6#F$$!3pJ?4+E<y[!#C/-
I\"yGF%F,$\"2gczxY#******!#</-I\"uGF%F,$!3\\[#3pG+++\"F5/-I\"vGF%F,$!3
Wqn[yn$[H\"!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "7'/I\"tG6\"$\"0z*e`EfT
J!#9/-I\"xGF%6#F$$!3![;g'zu(*****!#=/-I\"yGF%F,$!3g,*\\1t8>!R!#B/-I\"u
GF%F,$\"3qU;fo'))4z%F5/-I\"vGF%F,$!3UC\"yQH,++\"!#<" }}{PARA 11 "" 1 "
" {XPPMATH 20 "7'/I\"tG6\"$\"0o%Q!)*)Q7Z!#9/-I\"xGF%6#F$$\"3Gt#3y+Da5
\"!#A/-I\"yGF%F,$!3OFbe`e'*****!#=/-I\"uGF%F,$\"3[!>(o))>++5!#</-I\"vG
F%F,$\"3gKj8W4WQ6F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "7'/I\"tG6\"$\"0ez
rI&=$G'!#9/-I\"xGF%6#F$$\"3c9tfm;'*****!#=/-I\"yGF%F,$\"3#QS&e4G'3)>!#
A/-I\"uGF%F,$!3l#*4RV9[3?F5/-I\"vGF%F,$\"3)epnQ\">++5!#<" }}}{PARA 0 "
" 0 "" {TEXT 209 67 "We are really interested in the values of x and y
 at those 4 times." }{TEXT 209 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "P1a := [op(2,P1[2]),op(2,P1[3])];\n" }{MPLTEXT 1 0 
34 "P2a := [op(2,P2[2]),op(2,P2[3])];\n" }{MPLTEXT 1 0 34 "P3a := [op(
2,P3[2]),op(2,P3[3])];\n" }{MPLTEXT 1 0 33 "P4a := [op(2,P4[2]),op(2,P
4[3])];" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$$!3pJ?4+
E<y[!#C$\"2gczxY#******!#<" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$$!3![;g'
zu(*****!#=$!3g,*\\1t8>!R!#B" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$$\"3Gt
#3y+Da5\"!#A$!3OFbe`e'*****!#=" }}{PARA 11 "" 1 "" {XPPMATH 20 "7$$\"3
c9tfm;'*****!#=$\"3#QS&e4G'3)>!#A" }}}{PARA 0 "" 0 "" {TEXT 209 146 "T
his lets us see that with 6 decimal points of accuracy, the path is a \+
circle with period 2Pi.  We can graph the results to see this more cle
arly." }{TEXT 209 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "wit
h(plots):\n" }{MPLTEXT 1 0 69 "pointplot([seq([op(2,solcurve(t*.05)[2]
),op(2,solcurve(t*.05)[3])], \n" }{MPLTEXT 1 0 35 "         t=0..125)]
, connect=true);" }{MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {TEXT 217 0 ""
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$\"+&R<<g*!#5$!+#fpRz#!#57$$\"+6&e$H(*!#5$!+X&*e5B!#57$$\"+(=\"oK)*!#5
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217 0 "" }}}{PARA 0 "" 0 "" {TEXT 209 59 "We can also see what happens
 if we change the eccentricity." }{TEXT 209 0 "" }}{EXCHG {PARA 0 "> "
 0 "" {MPLTEXT 1 0 42 "e := 0.8; a := 1; tmax := 2*Pi*a*sqrt(a):\n" }
{MPLTEXT 1 0 24 "ICx := x(0) = a*(1-e); \n" }{MPLTEXT 1 0 17 "ICy := y
(0) = 0:\n" }{MPLTEXT 1 0 17 "ICu := u(0) = 0:\n" }{MPLTEXT 1 0 44 "IC
v := v(0) = evalf(sqrt((1+e)/(a*(1-e))));\n" }{MPLTEXT 1 0 66 "solcurv
e := dsolve(\{ODEa, ODEb, ODEc, ODEd, ICx, ICy, ICu, ICv\}, \n" }
{MPLTEXT 1 0 41 "     [x(t), y(t), u(t), v(t)], numeric);\n" }{MPLTEXT
 1 0 47 "pointplot([seq([op(2,solcurve(t*tmax/100)[2]),\n" }{MPLTEXT 
1 0 38 "     op(2,solcurve(t*tmax/100)[3])], \n" }{MPLTEXT 1 0 56 "   \+
      t=0..100)], connect=true, scaling=constrained);" }{MPLTEXT 1 0 
0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "$\"\")!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "/-I\"xG6\"6#\"\"!
$\"\"#!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "/-I\"vG6\"6#\"\"!$\"+++++
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$$!+,VVBL!#5$!+fxZ.`!#57$$!+Bht3C!#5$!*&)3X(\\!\"*7$$!+R:>Q9!#5$!+/JjF
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CALINGG6#%,CONSTRAINEDG" 1 2 2 1 10 1 2 6 0 4 1 1.0 45.0 45.0 0 0 "Cur
ve 1" }}{TEXT 217 0 "" }}}{PARA 0 "" 0 "" {TEXT 209 44 "We see that we
 get an ellipse for the orbit." }}{PARA 0 "" 0 "" {TEXT 209 0 "" }}
{PARA 0 "" 0 "" {TEXT 209 62 "We are now ready to return to the questi
on of relative orbits." }}{PARA 0 "" 0 "" {TEXT 209 0 "" }}{PARA 0 "" 
0 "" {TEXT 209 92 "Consider first the problem of the earth and Mercury
.  We will measure in astronomical units." }{TEXT 209 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "ee := 0.02; ae := 1; tmaxe := 2*Pi*
ae*sqrt(ae);\n" }{MPLTEXT 1 0 27 "ICxe := x(0) = ae*(1-ee); \n" }
{MPLTEXT 1 0 18 "ICye := y(0) = 0:\n" }{MPLTEXT 1 0 18 "ICue := u(0) =
 0:\n" }{MPLTEXT 1 0 48 "ICve := v(0) = evalf(sqrt((1+ee)/(ae*(1-ee)))
);\n" }{MPLTEXT 1 0 71 "solcurvee := dsolve(\{ODEa, ODEb, ODEc, ODEd, \+
ICxe, ICye, ICue, ICve\}, \n" }{MPLTEXT 1 0 41 "     [x(t), y(t), u(t)
, v(t)], numeric);\n" }{MPLTEXT 1 0 52 "em := 0.21; am := 36/93; tmaxm
 := 2*Pi*am*sqrt(am);\n" }{MPLTEXT 1 0 27 "ICxm := x(0) = am*(1-em); \+
\n" }{MPLTEXT 1 0 18 "ICym := y(0) = 0:\n" }{MPLTEXT 1 0 18 "ICum := u
(0) = 0:\n" }{MPLTEXT 1 0 48 "ICvm := v(0) = evalf(sqrt((1+em)/(am*(1-
em))));\n" }{MPLTEXT 1 0 71 "solcurvem := dsolve(\{ODEa, ODEb, ODEc, O
DEd, ICxm, ICym, ICum, ICvm\}, \n" }{MPLTEXT 1 0 40 "     [x(t), y(t),
 u(t), v(t)], numeric);" }{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "$\"\"#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "\"\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"\"#\"\"\"I#PiG%*protectedGF%F%" }}
{PARA 11 "" 1 "" {XPPMATH 20 "/-I\"xG6\"6#\"\"!$\"#)*!\"#" }}{PARA 11 
"" 1 "" {XPPMATH 20 "/-I\"vG6\"6#\"\"!$\"+hS??5!\"*" }}{PARA 11 "" 1 "
" {XPPMATH 20 "f*6#I(x_rkf45G6\"6)I$resGF%I%dataGF%I%varsGF%I)solnproc
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{XPPMATH 20 ",$*(#\"#[\"$h*\"\"\"I#PiG%*protectedGF')\"#$*#F'\"\"#F'F'
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{PARA 11 "" 1 "" {XPPMATH 20 "/-I\"vG6\"6#\"\"!$\"+y$e\"*)>!\"*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "f*6#I(x_rkf45G6\"6)I$resGF%I%dataGF%I%va
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{PARA 11 "" 1 "" {XPPMATH 20 ",$*&\"\"%\"\"\"I#PiG%*protectedGF%F%" }}
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}}}{PARA 0 "" 0 "" {TEXT 209 123 "Notice the slight change in the rela
tive path of the two planets.  Some of the inner loops are clearly sma
ller than others." }}{PARA 0 "" 0 "" {TEXT 209 0 "" }}{PARA 0 "" 0 "" 
{TEXT 209 48 "Now we try the same trick with Halley's comet.  " }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "ee := 0.02; ae := 1; tmaxe :
= 2*Pi*ae*sqrt(ae);\n" }{MPLTEXT 1 0 27 "ICxe := x(0) = ae*(1-ee); \n"
 }{MPLTEXT 1 0 18 "ICye := y(0) = 0:\n" }{MPLTEXT 1 0 18 "ICue := u(0)
 = 0:\n" }{MPLTEXT 1 0 48 "ICve := v(0) = evalf(sqrt((1+ee)/(ae*(1-ee)
)));\n" }{MPLTEXT 1 0 71 "solcurvee := dsolve(\{ODEa, ODEb, ODEc, ODEd
, ICxe, ICye, ICue, ICve\}, \n" }{MPLTEXT 1 0 41 "     [x(t), y(t), u(
t), v(t)], numeric);\n" }{MPLTEXT 1 0 61 "em := 0.97; am := 1680/93; t
maxm := evalf(2*Pi*am*sqrt(am));\n" }{MPLTEXT 1 0 27 "ICxm := x(0) = a
m*(1-em); \n" }{MPLTEXT 1 0 18 "ICym := y(0) = 0:\n" }{MPLTEXT 1 0 18 
"ICum := u(0) = 0:\n" }{MPLTEXT 1 0 48 "ICvm := v(0) = evalf(sqrt((1+e
m)/(am*(1-em))));\n" }{MPLTEXT 1 0 71 "solcurvem := dsolve(\{ODEa, ODE
b, ODEc, ODEd, ICxm, ICym, ICum, ICvm\}, \n" }{MPLTEXT 1 0 41 "     [x
(t), y(t), u(t), v(t)], numeric);\n" }{MPLTEXT 1 0 29 "tmax := 2*max(t
maxe, tmaxm);\n" }{MPLTEXT 1 0 16 "pointplot([seq(\n" }{MPLTEXT 1 0 
66 "[op(2,solcurvee(t*tmax/1000)[2])-op(2,solcurvem(t*tmax/1000)[2]),
\n" }{MPLTEXT 1 0 66 "op(2,solcurvee(t*tmax/1000)[3])-op(2,solcurvem(t
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things to notice from the computations." }{TEXT 209 0 "" }}{PARA 0 "" 
0 "" {TEXT 209 0 "" }}{PARA 0 "" 0 "" {TEXT 209 97 "At its closest poi
nt, Halley's comet is inside the earth's orbit and moving almost twice
 as fast." }}{PARA 0 "" 0 "" {TEXT 209 80 "If a year is 2Pi, then Hall
ey's comet orbits once every (482/(2Pi) = 76.7 years." }{TEXT 209 0 ""
 }}{PARA 0 "" 0 "" {TEXT 209 0 "" }}{PARA 0 "" 0 "" {TEXT 209 66 "The \+
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