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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 21 "CRTB Invariant Curves" }}{PARA 
19 "" 0 "" {TEXT -1 15 "Marc A. Murison" }}{PARA 256 "" 0 "" {TEXT -1 
59 "Astronomical Applications Department\nU.S. Naval Observatory" }}
{PARA 258 "" 0 "" {TEXT -1 57 "murison@aa.usno.navy.mil\nhttp://aa.usn
o.navy.mil/murison/" }}{PARA 257 "" 0 "" {TEXT -1 13 "9 March, 1998" }
}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 12 "Introduction" }}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "restart;" "6#%(restartG" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 75 "The Jacobi constant of the circular restricted three-b
ody (CRTB) problem is" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "Omega = (xi^2+
eta^2+zeta^2)/2+(1-mu)/r[1]+mu/r[2]:" "6#/%&OmegaG,(*&,(*$%#xiG\"\"#\"
\"\"*$%$etaG\"\"#F+*$%%zetaG\"\"#F+F+\"\"#!\"\"F+*&,&\"\"\"F+%#muGF3F+
&%\"rG6#\"\"\"F3F+*&F7F+&F96#\"\"#F3F+" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "JC := %:" "6#>%#JCG%\"%G" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 6 "where " }{XPPEDIT 18 0 "mu = m[2]/(m[1]+m[2])" "6#/%#muG*&
&%\"mG6#\"\"#\"\"\",&&F'6#\"\"\"F*&F'6#\"\"#F*!\"\"" }{TEXT -1 2 ", " 
}{XPPEDIT 19 1 "r[1] := sqrt((xi+mu)^2+eta^2+zeta^2):" "6#>&%\"rG6#\"
\"\"-%%sqrtG6#,(*$,&%#xiG\"\"\"%#muGF/\"\"#F/*$%$etaG\"\"#F/*$%%zetaG
\"\"#F/" }{TEXT -1 6 ", and " }{MPLTEXT 0 21 0 "" }{XPPEDIT 19 1 "r[2]
 := sqrt((xi-(1-mu))^2+eta^2+zeta^2):" "6#>&%\"rG6#\"\"#-%%sqrtG6#,(*$
,&%#xiG\"\"\",&\"\"\"F/%#muG!\"\"F3\"\"#F/*$%$etaG\"\"#F/*$%%zetaG\"\"
#F/" }{TEXT -1 20 ". The coordinates ( " }{XPPEDIT 18 0 "xi,eta,zeta;
" "6%%#xiG%$etaG%%zetaG" }{TEXT -1 85 " ) are in the rotating frame.  \+
(Historically, the Jacobi constant is usually denoted " }{XPPEDIT 18 
0 "C = 2*Omega;" "6#/%\"CG*&\"\"#\"\"\"%&OmegaGF'" }{TEXT -1 2 ".)" }}
}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "Surface Plot \227 (x,y)" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Here is a procedure to evaluate th
e Jacobi constant:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "S := fn(r
hs(JC),mu,xi,eta,zeta);" "6#>%\"SG-%#fnG6'-%$rhsG6#%#JCG%#muG%#xiG%$et
aG%%zetaG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"SGR6&%#muG%#xiG%$etaG
%%zetaG6\"6$%)operatorG%&arrowGF+,,*$)9%\"\"#\"\"\"#\"\"\"F3*$)9&F3F4F
5*$)9'F3F4F5*&,&F6F69$!\"\"F4*$-%%sqrtG6#,,F0F6*&F2F6F?F6F3*$)F?F3F4F6
F7F6F:F6F4!\"\"F6*&F?F4*$-FC6#,2F0F6F2!\"#FFF3F6F6F?FOFGF6F7F6F:F6F4FI
F6F+F+F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "For plotting, the fol
lowing two procedures are useful:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
19 1 "C := proc( Cmin, mu, xi, eta, zeta )\n    local jc;\n    jc := -
S( mu, xi, eta, zeta );\n    if( jc < Cmin ) then jc := Cmin; fi;\n   \+
 jc;\nend:" "6#>%\"CGR6'%%CminG%#muG%#xiG%$etaG%%zetaG7#%#jcG6\"F.C%>F
-,$-%\"SG6&F(F)F*F+!\"\"@$2F-F'>F-F'F-F.F.F." }}}{EXCHG {PARA 0 "" 0 "
" {XPPEDIT 19 1 "Cplot := proc( mu, z, xmin, xmax, ymin, ymax, Cmin, \+
\n               n::posint, g::list, doscale )\n    local xy_scale, ax
istype, styletype, plt3D, plt2D, f,\n          xform, disp, L, Cmax, c
ons, k;\n    if doscale then \n        xy_scale  := Cmin/(xmax - xmin)
; \n        axistype  := NONE;\n        styletype := PATCHNOGRID;\n   \+
 else \n        xy_scale  := 1; \n        axistype  := FRAMED;\n      \+
  styletype := PATCHCONTOUR;\n    fi;\n    L := light4;\n    Cmax := -
2.8;\n    cons := [ seq(Cmin + k/n*(Cmax-Cmin), k=0..n-1) ]; \n    plt
3D := plot3d( \n       [xy_scale*x, xy_scale*y, 2*'C'(Cmin/2,mu,x,y,z)
], \n        x=xmin..xmax, y=ymin..ymax, contours=cons,\n        light
model=L, axes=axistype, grid='g', style=styletype );\n    plt2D := plo
ts[contourplot](\n        2*'C'(Cmin/2,mu,x,y,z), x=xmin..xmax, y=ymin
..ymax, \n        grid='g', contours=cons, filled=true );\n#   xform :
= proc( x, y ) global Cmin; [x,y,Cmin]; end:\n    f := plottools[trans
form]((x,y)->[x,y,Cmin]):\n    if doscale then\n        plt3D;\n    el
se\n        plots[display]( \{ plt3D, f(plt2D) \}, \n            orien
tation=[-65,60], projection=ORTHOGONAL, \n            axes=axistype, l
ightmodel=L, \n            labels=[`x`,`y`,`C`] );\n    fi;\nend:" "6#
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-\"\"\",&F*FHF)!\"\"FJ>F7%%NONEG>F8%,PATCHNOGRIDGC%>F6\"\"\">F7%'FRAME
DG>F8%-PATCHCONTOURG>F>%'light4G>F?,$$\"#G!\"\"FJ>F@7#-%$seqG6$,&F-FH*
(FAFHF/FJ,&F?FHF-FJFHFH/FA;\"\"!,&F/FH\"\"\"FJ>F9-%'plot3dG6*7%*&F6FH%
\"xGFH*&F6FH%\"yGFH*&\"\"#FH-.%\"CG6'*&F-FH\"\"#FJF'FjoF\\pF(FH/Fjo;F)
F*/F\\p;F+F,/%)contoursGF@/%+lightmodelGF>/%%axesGF7/%%gridG.F2/%&styl
eGF8>F:-&%&plotsG6#%,contourplotG6(*&\"\"#FH-.Fap6'*&F-FH\"\"#FJF'FjoF
\\pF(FH/Fjo;F)F*/F\\p;F+F,/F`q.F2/FjpF@/%'filledG%%trueG>F;-&%*plottoo
lsG6#%*transformG6#R6$FjoF\\p7\"6$%)operatorG%&arrowGFB7%FjoF\\pF-FBFB
FB@%F4F9-&Fgq6#%(displayG6(<$F9-F;6#F:/%,orientationG7$,$\"#lFJ\"#g/%+
projectionG%+ORTHOGONALG/F^qF7/F\\qF>/%'labelsG7%FjoF\\pFapFBFBFB" }}
{PARA 7 "" 1 "" {TEXT -1 47 "Warning, `Cmin` is a lexically scoped par
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acobi surface for " }{XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT -1 19 " = 0.
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6#%.UNCONSTRAINEDG-F_\\fm6#%&PATCHG-%+LIGHTMODELG6#%(LIGHT_4G-%+AXESLA
BELSG6&%\"xG%\"yG%\"CG-%%FONTG6%%&TIMESG%&ROMANGF^ben-%*AXESTICKSG6&%(
DEFAULTGFhb[oFhb[o-Fab[o6$%*HELVETICAGFdjgn-%&TITLEG6$%!G-Fab[o6%Fcb[o
%'ITALICGF^ben-%)CONTOURSG6#7;$F-F^cn$!++++_()F-$!++++/&)F-$!++++c#)F-
$!++++3!)F-$!++++gxF-$!++++7vF-$!++++ksF-$!++++;qF-$!++++onF-$!++++?lF
-$!++++siF-$!++++CgF-$!++++wdF-$!++++GbF-$!++++!G&F-$!++++K]F-$!++++%y
%F-$!++++OXF-$!++++)G%F-$!++++SSF-$!++++#z$F-$!++++WNF-$!++++'H$F-$!++
++[IF--%*AXESSTYLEG6#%&FRAMEG-%+PROJECTIONG6%$!#lF^cn$\"#gF^cnF]\\fm" 
1 2 0 1 0 2 4 6 3 2 1.000000 60.000000 -65.000000 1 }}}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 122 "Let's make a VRML object of the surface, suita
ble for viewing in a VRML browser such as Cosmo (see http://cosmo.sgi.
com/):" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "plottools[vrml](Cplot
(.15,0,-1.3,1.5,-1.4,1.4,-9,25,[35, 35],true),\"d:/dynamics/ertb/CRTBS
urfaceXY.wrl\",background_color = black,shininess = 1,smooth_shading =
 true);" "6#-&%*plottoolsG6#%%vrmlG6'-%&CplotG6,$\"#:!\"#\"\"!,$$\"#8!
\"\"!\"\"$\"#:!\"\",$$\"#9!\"\"F4$\"#9!\"\",$\"\"*F4\"#D7$\"#N\"#N%%tr
ueGQCd:/dynamics/ertb/CRTBSurfaceXY.wrl6\"/%1background_colorG%&blackG
/%*shininessG\"\"\"/%/smooth_shadingGFE" }}}}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 18 "Surface Plot \227 (R," }{XPPEDIT 257 0 "theta;" "6#%&thet
aG" }{TEXT -1 1 ")" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "Define " }
{XPPEDIT 18 0 "R = sqrt(xi^2+eta^2+zeta^2);" "6#/%\"RG-%%sqrtG6#,(*$%#
xiG\"\"#\"\"\"*$%$etaG\"\"#F,*$%%zetaG\"\"#F," }{TEXT -1 44 " and conv
ert to spherical polar coordinates." }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "Omega = map(simplify,rhs(JC),\{xi^2+eta^2+zeta^2 = R^2
\}):" "6#/%&OmegaG-%$mapG6%%)simplifyG-%$rhsG6#%#JCG<#/,(*$%#xiG\"\"#
\"\"\"*$%$etaG\"\"#F3*$%%zetaG\"\"#F3*$%\"RG\"\"#" }}{PARA 0 "" 0 "" 
{XPPEDIT 19 1 "subs(xi = R*cos(phi)*sin(theta),%);" "6#-%%subsG6$/%#xi
G*(%\"RG\"\"\"-%$cosG6#%$phiGF*-%$sinG6#%&thetaGF*%\"%G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/%&OmegaG,(*$)%\"RG\"\"#\"\"\"#\"\"\"F)*&,&!\"\"
F,%#muGF,F**$-%%sqrtG6#,(**F(F,-%$cosG6#%$phiGF,-%$sinG6#%&thetaGF,F0F
,F)*$)F0F)F*F,F&F,F*!\"\"F/*&F0F**$-F36#,,F?F,F0!\"#F,F,**,&FGF,F0F)F,
F(F*F7F*F;F*F,F&F,F*FAF," }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "JCp
olar := %:" "6#>%(JCpolarG%\"%G" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
19 1 "Spolar := fn(rhs(JCpolar), mu, R, theta, phi);" "6#>%'SpolarG-%#
fnG6'-%$rhsG6#%(JCpolarG%#muG%\"RG%&thetaG%$phiG" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%'SpolarGR6&%#muG%\"RG%&thetaG%$phiG6\"6$%)operatorG%&
arrowGF+,(*$)9%\"\"#\"\"\"#\"\"\"F3*&,&!\"\"F69$F6F4*$-%%sqrtG6#,(**F2
F6-%$cosG6#9'F6-%$sinG6#9&F6F:F6F3*$)F:F3F4F6F0F6F4!\"\"F9*&F:F4*$-F=6
#,,FIF6F:!\"#F6F6**,&FQF6F:F3F6F2F4FAF4FEF4F6F0F6F4FKF6F+F+F+" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Cpolar := proc (Cmin, mu, R, the
ta, phi) local jc; jc := -Spolar(mu,R,theta,phi); if jc < Cmin then jc
 := Cmin fi; jc end:" "6#>%'CpolarGR6'%%CminG%#muG%\"RG%&thetaG%$phiG7
#%#jcG6\"F.C%>F-,$-%'SpolarG6&F(F)F*F+!\"\"@$2F-F'>F-F'F-F.F.F." }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "readlib(addcoords)(z_cylindrical
,[z,r,phi],[r*cos(phi),r*sin(phi),z]):" "6#--%(readlibG6#%*addcoordsG6
%%.z_cylindricalG7%%\"zG%\"rG%$phiG7%*&F,\"\"\"-%$cosG6#F-F0*&F,F0-%$s
inG6#F-F0F+" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Cpolarplot := pr
oc ( F::procedure, mu, theta, Rmin, Rmax, Cmin, \n                    \+
 n::posint, g::list, doscale ) \n    local Rscale, axistype, styletype
, plt3D, plt2D, f, xform, disp, \n          L, cons, Cmax, k; \n    if
 doscale then \n        Rscale    := 1/2*Cmin/(Rmax-Rmin); \n        a
xistype  := NONE; \n        styletype := PATCHNOGRID \n    else \n    \+
    Rscale    := 1; \n        axistype  := FRAMED; \n        styletype
 := PATCHCONTOUR \n    fi; \n    L     := light4; \n    Cmax  := -2.8;
 \n    cons  := [seq(Cmin+k/n*(Cmax-Cmin),k = 0 .. n-1)]; \n    plt3D \+
:= plot3d( [2*'F'(Cmin/2,mu,R,theta,phi),R*Rscale,phi],\n        R = R
min .. Rmax, phi = 0 .. 2*Pi, coords = z_cylindrical,\n        contour
s = cons, lightmodel = L, axes = axistype, \n        grid = ('g'), sty
le = styletype); \n    plt2D := plots[contourplot]( [2*'F'(Cmin/2,mu,R
,theta,phi),R*Rscale,phi],\n        R = Rmin .. Rmax, phi = 0 .. 2*Pi,
 coords = z_cylindrical,\n        contours = cons, filled = true); \n \+
   f := plottools[transform](proc (x, y) options operator, arrow; [x, \+
y, Cmin] end); \n    if doscale then \n        plt3D \n    else \n    \+
    plots[display](\{plt3D, f(plt2D)\},\n            orientation = [-6
5, 60], projection = ORTHOGONAL,\n            axes = axistype, lightmo
del = L, labels = [x, y, C]) \n    fi \nend:" "6#>%+CpolarplotGR6+'%\"
FG%*procedureG%#muG%&thetaG%%RminG%%RmaxG%%CminG'%\"nG%'posintG'%\"gG%
%listG%(doscaleG7.%'RscaleG%)axistypeG%*styletypeG%&plt3DG%&plt2DG%\"f
G%&xformG%%dispG%\"LG%%consG%%CmaxG%\"kG6\"FCC*@%F5C%>F7**\"\"\"\"\"\"
\"\"#!\"\"F.FJ,&F-FJF,FLFL>F8%%NONEG>F9%,PATCHNOGRIDGC%>F7\"\"\">F8%'F
RAMEDG>F9%-PATCHCONTOURG>F?%'light4G>FA,$$\"#G!\"\"FL>F@7#-%$seqG6$,&F
.FJ*(FBFJF0FL,&FAFJF.FLFJFJ/FB;\"\"!,&F0FJ\"\"\"FL>F:-%'plot3dG6+7%*&
\"\"#FJ-.F(6'*&F.FJ\"\"#FLF*%\"RGF+%$phiGFJ*&FcpFJF7FJFdp/Fcp;F,F-/Fdp
;Fdo*&\"\"#FJ%#PiGFJ/%'coordsG%.z_cylindricalG/%)contoursGF@/%+lightmo
delGF?/%%axesGF8/%%gridG.F3/%&styleGF9>F;-&%&plotsG6#%,contourplotG6(7
%*&\"\"#FJ-.F(6'*&F.FJ\"\"#FLF*FcpF+FdpFJ*&FcpFJF7FJFdp/Fcp;F,F-/Fdp;F
do*&\"\"#FJF\\qFJ/F^qF_q/FaqF@/%'filledG%%trueG>F<-&%*plottoolsG6#%*tr
ansformG6#R6$%\"xG%\"yG7\"6$%)operatorG%&arrowGFC7%F_tF`tF.FCFCFC@%F5F
:-&F^r6#%(displayG6(<$F:-F<6#F;/%,orientationG7$,$\"#lFL\"#g/%+project
ionG%+ORTHOGONALG/FeqF8/FcqF?/%'labelsG7%F_tF`t%\"CGFCFCFC" }}{PARA 7 
"" 1 "" {TEXT -1 51 "Warning, `\\`Cmin\\` is a lexically scoped parame
ter`" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Here is a plot of the Jac
obi surface for " }{XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT -1 41 " = 0.15
 and inclination angle 0 degrees: " }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "Cpolarplot(Cpolar,.15,Pi/2,0,1.6,-8,30,[30, 120],false)
;" "6#-%+CpolarplotG6+%'CpolarG$\"#:!\"#*&%#PiG\"\"\"\"\"#!\"\"\"\"!$
\"#;!\"\",$\"\")F.\"#I7$\"#I\"$?\"%&falseG" }}{PARA 13 "" 1 "" 
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