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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 73 "Determination of Two-Body
 Orbital Elements from the Two-Body State Vector" }}{PARA 19 "" 0 "" 
{TEXT -1 15 "Marc A. Murison" }}{PARA 256 "" 0 "" {TEXT -1 74 "Astrono
mical Applications Department\nU.S. Naval Observatory\nWashington, DC
" }}{PARA 258 "" 0 "" {TEXT -1 57 "murison@aa.usno.navy.mil\nhttp://aa
.usno.navy.mil/murison/" }}{PARA 257 "" 0 "" {TEXT -1 16 "October 23, \+
1998" }}}{SECT 0 {PARA 3 "" 0 "1. Initialization" {TEXT -1 17 "1. Init
ialization" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "restart;" "6#%(res
tartG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "alias(r = r(t),r[e] = \+
r[e](t),r[h] = r[h](t),v[r] = diff(r(t),t),v[r[e]] = diff(r[e](t),t),v
[r[h]] = diff(r[h](t),t),theta = theta(t),x = x(t),y = y(t),z = z(t),v
[x] = diff(x(t),t),v[y] = diff(y(t),t),v[z] = diff(z(t),t)):" "6#-%&al
iasG6//%\"rG-F'6#%\"tG/&F'6#%\"eG-&F'6#F.6#F*/&F'6#%\"hG-&F'6#F66#F*/&
%\"vG6#F'-%%diffG6$-F'6#F*F*/&F=6#&F'6#F.-F@6$-&F'6#F.6#F*F*/&F=6#&F'6
#F6-F@6$-&F'6#F66#F*F*/%&thetaG-Fen6#F*/%\"xG-Fin6#F*/%\"yG-F]o6#F*/%
\"zG-Fao6#F*/&F=6#Fin-F@6$-Fin6#F*F*/&F=6#F]o-F@6$-F]o6#F*F*/&F=6#Fao-
F@6$-Fao6#F*F*" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 73 "a utility proc
edure to convert a vector equation into component equations" }}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "mat2eqs := proc( Meqn::equation )\n  lo
cal k, eqlist, Mleft, Mright;\n  Mleft  := evalm(lhs(Meqn));\n  Mright
 := evalm(rhs(Meqn));\n  eqlist := [];\n  for k from 1 to rows(Mleft) \+
do\n    eqlist := [ op(eqlist), Mleft[k,1]=Mright[k,1] ];\n  od;\n  ev
al(eqlist);\nend:" "6#>%(mat2eqsGR6#'%%MeqnG%)equationG7&%\"kG%'eqlist
G%&MleftG%'MrightG6\"F/C'>F--%&evalmG6#-%$lhsG6#F(>F.-F36#-%$rhsG6#F(>
F,7\"?(F+\"\"\"\"\"\"-%%rowsG6#F-%%trueG>F,7$-%#opG6#F,/&F-6$F+\"\"\"&
F.6$F+\"\"\"-%%evalG6#F,F/F/F/" }}}}}{SECT 0 {PARA 3 "" 0 "2. Two-Body
 Motion Definitions and Derivations" {TEXT -1 46 "2. Two-Body Motion D
efinitions and Derivations" }}{SECT 0 {PARA 4 "" 0 "2.1. angular momen
tum" {TEXT -1 21 "2.1. angular momentum" }}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "matrix([[h[x]], [h[y]], [h[z]]]) = mat(cross([x, y, z],
[v[x], v[y], v[z]]));" "6#/-%'matrixG6#7%7#&%\"hG6#%\"xG7#&F*6#%\"yG7#
&F*6#%\"zG-%$matG6#-%&crossG6$7%F,F0F47%&%\"vG6#F,&F>6#F0&F>6#F4" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"hG6#%\"xG7#&F*6#%
\"yG7#&F*6#%\"zG-F%6#7%7#,&*&F0\"\"\"&%\"vG6#F4F;F;*&F4F;&F=6#F0F;!\"
\"7#,&*&F4\"\"\"&F=6#F,F;F;*&F,F;F<FFFB7#,&*&F,FFF@FFF;*&F0FFFGFFFB" }
}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "hvec := %:" "6#>%%hvecG%\"%G" 
}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "h = sqrt(dot(rhs(hvec),rhs(hv
ec)));" "6#/%\"hG-%%sqrtG6#-%$dotG6$-%$rhsG6#%%hvecG-F,6#F." }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/%\"hG*$-%%sqrtG6#,(*$),&*&%\"yG\"\"\"&%\"vG
6#%\"zGF/F/*&F3F/&F16#F.F/!\"\"\"\"#\"\"\"F/*$),&*&F3F9&F16#%\"xGF/F/*
&F@F/F0F9F7F8F9F/*$),&*&F@F9F5F9F/*&F.F9F>F9F7F8F9F/F9" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 30 "For elliptical motion, we have" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "h = sqrt(lambda*a*(1-e^2)):" "6#
/%\"hG-%%sqrtG6#*(%'lambdaG\"\"\"%\"aGF*,&\"\"\"F**$%\"eG\"\"#!\"\"F*
" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "hesub := %:" "6#>%&hesubG%
\"%G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "la
mbda = G*(m[1]+m[2]);" "6#/%'lambdaG*&%\"GG\"\"\",&&%\"mG6#\"\"\"F'&F*
6#\"\"#F'F'" }{TEXT -1 25 ".  For hyperbolic motion," }}}{EXCHG {PARA 
0 "" 0 "" {XPPEDIT 19 1 "h = sqrt(lambda*a*(e^2-1)):" "6#/%\"hG-%%sqrt
G6#*(%'lambdaG\"\"\"%\"aGF*,&*$%\"eG\"\"#F*\"\"\"!\"\"F*" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "hhsub := %:" "6#>%&hhsubG%\"%G" }}}}
{SECT 0 {PARA 4 "" 0 "2.2. radius" {TEXT -1 11 "2.2. radius" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 11 "Elliptical:" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "r = a*(1-e^2)/(1+e*cos(theta)):" "6#/%\"rG*(%\"aG\"\"\"
,&\"\"\"F'*$%\"eG\"\"#!\"\"F',&\"\"\"F'*&F+F'-%$cosG6#%&thetaGF'F'F-" 
}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "resub := %:" "6#>%&resubG%\"%
G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 11 "Hyperbolic:" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "r = a*(e^2-1)/(1+e*cos(theta)):" "6#/%
\"rG*(%\"aG\"\"\",&*$%\"eG\"\"#F'\"\"\"!\"\"F',&\"\"\"F'*&F*F'-%$cosG6
#%&thetaGF'F'F-" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "rhsub := %:
" "6#>%&rhsubG%\"%G" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "matrix([
[x], [y], [z]]) = r*matrix([[cos(Omega)*cos(omega+theta)-sin(Omega)*si
n(omega+theta)*cos(i)], [sin(Omega)*cos(omega+theta)+cos(Omega)*sin(om
ega+theta)*cos(i)], [sin(omega+theta)*sin(i)]]):" "6#/-%'matrixG6#7%7#
%\"xG7#%\"yG7#%\"zG*&%\"rG\"\"\"-F%6#7%7#,&*&-%$cosG6#%&OmegaGF0-F86#,
&%&omegaGF0%&thetaGF0F0F0*(-%$sinG6#F:F0-FB6#,&F>F0F?F0F0-F86#%\"iGF0!
\"\"7#,&*&-FB6#F:F0-F86#,&F>F0F?F0F0F0*(-F86#F:F0-FB6#,&F>F0F?F0F0-F86
#FIF0F07#*&-FB6#,&F>F0F?F0F0-FB6#FIF0F0" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "rvec := %:" "6#>%%rvecG%\"%G" }}}{EXCHG {PARA 0 "" 0 "
" {XPPEDIT 19 1 "mat2eqs(rvec);" "6#-%(mat2eqsG6#%%rvecG" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#7%/%\"xG*&%\"rG\"\"\",&*&-%$cosG6#%&OmegaGF(-F,6
#,&%&omegaGF(%&thetaGF(F(F(*(-%$sinGF-F(-F6F0F(-F,6#%\"iGF(!\"\"F(/%\"
yG*&F'\"\"\",&*&F5F?F/F?F(*(F+F?F7F?F8F?F(F(/%\"zG*(F'F?F7F?-F6F9F(" }
}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "reqs := %:" "6#>%%reqsG%\"%G" 
}}}}{SECT 0 {PARA 4 "" 0 "2.3. velocity" {TEXT -1 13 "2.3. velocity" }
}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 35 "radial velocity \227 elliptical \+
motion" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "diff(resub,t);" "6#-%%
diffG6$%&resubG%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"vG6#%\"rG
*&*,%\"aG\"\"\",&F+F+*$)%\"eG\"\"#\"\"\"!\"\"F+F/F1-%$sinG6#%&thetaGF+
-%%diffG6$F6%\"tGF+F1*$),&F+F+*&F/F+-%$cosGF5F+F+\"\"#F1!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(diff(theta,t) = h/(r^2),%);
" "6#-%%subsG6$/-%%diffG6$%&thetaG%\"tG*&%\"hG\"\"\"*$%\"rG\"\"#!\"\"%
\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"vG6#%\"rG*&*,%\"aG\"\"\",
&F+F+*$)%\"eG\"\"#\"\"\"!\"\"F+F/F1-%$sinG6#%&thetaGF+%\"hGF+F1*&),&F+
F+*&F/F+-%$cosGF5F+F+\"\"#F1)F'\"\"#F1!\"\"" }}}{EXCHG {PARA 0 "" 0 "
" {XPPEDIT 19 1 "lhs(%) = subs(resub,rhs(%));" "6#/-%$lhsG6#%\"%G-%%su
bsG6$%&resubG-%$rhsG6#F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"vG6#%
\"rG*&*(%\"eG\"\"\"-%$sinG6#%&thetaGF+%\"hGF+\"\"\"*&%\"aG\"\"\",&F+F+
*$)F*\"\"#F1!\"\"\"\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 
"subs(hesub,%);" "6#-%%subsG6$%&hesubG%\"%G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%\"vG6#%\"rG*&*(%\"eG\"\"\"-%$sinG6#%&thetaGF+-%%sqrt
G6#*(%'lambdaGF+%\"aGF+,&F+F+*$)F*\"\"#\"\"\"!\"\"F+F:F:*&F5\"\"\"F6\"
\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "vresub := %:" "6#>
%'vresubG%\"%G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 35 "radial veloci
ty \227 hyperbolic motion" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "dif
f(rhsub,t);" "6#-%%diffG6$%&rhsubG%\"tG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/&%\"vG6#%\"rG*&*,%\"aG\"\"\",&!\"\"F+*$)%\"eG\"\"#\"\"\"F+F+F0F
2-%$sinG6#%&thetaGF+-%%diffG6$F6%\"tGF+F2*$),&F+F+*&F0F+-%$cosGF5F+F+
\"\"#F2!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(diff(theta
,t) = h/(r^2),%);" "6#-%%subsG6$/-%%diffG6$%&thetaG%\"tG*&%\"hG\"\"\"*
$%\"rG\"\"#!\"\"%\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"vG6#%\"r
G*&*,%\"aG\"\"\",&!\"\"F+*$)%\"eG\"\"#\"\"\"F+F+F0F2-%$sinG6#%&thetaGF
+%\"hGF+F2*&),&F+F+*&F0F+-%$cosGF5F+F+\"\"#F2)F'\"\"#F2!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "lhs(%) = subs(rhsub,rhs(%));" "6
#/-%$lhsG6#%\"%G-%%subsG6$%&rhsubG-%$rhsG6#F'" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%\"vG6#%\"rG*&*(%\"eG\"\"\"-%$sinG6#%&thetaGF+%\"hGF+
\"\"\"*&%\"aG\"\"\",&!\"\"F+*$)F*\"\"#F1F+\"\"\"!\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(hhsub,%);" "6#-%%subsG6$%&hhsubG%
\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"vG6#%\"rG*&*(%\"eG\"\"\"-
%$sinG6#%&thetaGF+-%%sqrtG6#*(%'lambdaGF+%\"aGF+,&!\"\"F+*$)F*\"\"#\"
\"\"F+F+F;F;*&F5\"\"\"F6\"\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "vrhsub := %:" "6#>%'vrhsubG%\"%G" }}}}{SECT 0 {PARA 5 "
" 0 "" {TEXT -1 15 "velocity vector" }}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "map(diff,lhs(rvec),t) = map(diff,evalm(rhs(rvec)),t);" 
"6#/-%$mapG6%%%diffG-%$lhsG6#%%rvecG%\"tG-F%6%F'-%&evalmG6#-%$rhsG6#F+
F," }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"vG6#%\"xG7#
&F*6#%\"yG7#&F*6#%\"zG-F%6#7%7#,&*&&F*6#%\"rG\"\"\",&*&-%$cosG6#%&Omeg
aGF>-FB6#,&%&omegaGF>%&thetaGF>F>F>*(-%$sinGFCF>-FLFFF>-FB6#%\"iGF>!\"
\"F>F>*&F=F>,&*(FA\"\"\"FMFU-%%diffG6$FI%\"tGF>FQ**FKFUFEFUFVFUFNFUFQF
>F>7#,&*&F;FU,&*&FKFUFEFUF>*(FAFUFMFUFNFUF>F>F>*&F=FU,&*(FKFUFMFUFVFUF
Q**FAFUFEFUFVFUFNFUF>F>F>7#,&*(F;FUFMFU-FLFOF>F>**F=FUFEFUFVFUFboFUF>
" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Collect(%,[sin(i), r, v[r],
 diff(theta,t)]);" "6#-%(CollectG6$%\"%G7&-%$sinG6#%\"iG%\"rG&%\"vG6#F
,-%%diffG6$%&thetaG%\"tG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%'matrix
G6#7%7#&%\"vG6#%\"xG7#&F*6#%\"yG7#&F*6#%\"zG-F%6#7%7#,&*(,&*&-%$cosG6#
%&OmegaG\"\"\"-%$sinG6#,&%&omegaGFA%&thetaGFAFA!\"\"*(-FCF?FA-F>FDFA-F
>6#%\"iGFAFHFA-%%diffG6$FG%\"tGFA%\"rGFAFA*&&F*6#FSFA,&*&F=\"\"\"FKFYF
A*(FJFYFBFYFLFYFHFAFA7#,&*(,&*&FJFYFBFYFH*(F=FYFKFYFLFYFAFAFOFYFSFYFA*
&FUFY,&*&FJFYFKFYFA*(F=FYFBFYFLFYFAFAFA7#*&,&*&FUFYFBFYFA*(FSFYFKFYFOF
YFAFA-FCFMFA" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(diff(theta
,t) = h/(r^2),%);" "6#-%%subsG6$/-%%diffG6$%&thetaG%\"tG*&%\"hG\"\"\"*
$%\"rG\"\"#!\"\"%\"%G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#
7%7#&%\"vG6#%\"xG7#&F*6#%\"yG7#&F*6#%\"zG-F%6#7%7#,&*&*&,&*&-%$cosG6#%
&OmegaG\"\"\"-%$sinG6#,&%&omegaGFB%&thetaGFBFB!\"\"*(-FDF@FB-F?FEFB-F?
6#%\"iGFBFIFB%\"hGFB\"\"\"%\"rG!\"\"FB*&&F*6#FRFB,&*&F>FQFLFQFB*(FKFQF
CFQFMFQFIFBFB7#,&*&*&,&*&FKFQFCFQFI*(F>FQFLFQFMFQFBFBFPFQFQFRFSFB*&FUF
Q,&*&FKFQFLFQFB*(F>FQFCFQFMFQFBFBFB7#*&,&*&FUFQFCFQFB*&*&FLFQFPFQFQFRF
SFBFB-FDFNFB" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "vvec := %:" "6#
>%%vvecG%\"%G" }}}}}{SECT 0 {PARA 4 "" 0 "2.4. angular momentum revisi
ted" {TEXT -1 31 "2.4. angular momentum revisited" }}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 24 "The angular momentum is " }{XPPEDIT 18 0 "h = r*x*
v;" "6#/%\"hG*(%\"rG\"\"\"%\"xGF'%\"vGF'" }{TEXT -1 9 ".  Hence," }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "matrix([[h[x]], [h[y]], [h[z]]])
 = mat(cross(convert(evalm(rhs(rvec)),vector),convert(rhs(vvec),vector
)));" "6#/-%'matrixG6#7%7#&%\"hG6#%\"xG7#&F*6#%\"yG7#&F*6#%\"zG-%$matG
6#-%&crossG6$-%(convertG6$-%&evalmG6#-%$rhsG6#%%rvecG%'vectorG-F<6$-FB
6#%%vvecGFE" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"hG
6#%\"xG7#&F*6#%\"yG7#&F*6#%\"zG-F%6#7%7#,&**%\"rG\"\"\",&*&-%$sinG6#%&
OmegaGF<-%$cosG6#,&%&omegaGF<%&thetaGF<F<F<*(-FDFAF<-F@FEF<-FD6#%\"iGF
<F<F<,&*&&%\"vG6#F;F<FK\"\"\"F<*&*&FCFTF*F<FTF;!\"\"F<F<-F@FMF<F<**F;F
TFKFTFXFT,&*&*&,&*&F?FTFKFT!\"\"*(FJFTFCFTFLFTF<F<F*FTFTF;FWF<*&FQFTF=
FTF<F<Fin7#,&**F;FTFKFTFXFT,&*&*&,&*&FJFTFKFTFin*(F?FTFCFTFLFTFinF<F*F
TFTF;FWF<*&FQFT,&*&FJFTFCFTF<*(F?FTFKFTFLFTFinF<F<F<F<**F;FTFfoFTFOFTF
XFTFin7#,&*(F;FTFfoFTFZFTF<*(F;FTF=FTF_oFTFin" }}}{EXCHG {PARA 0 "" 0 
"" {XPPEDIT 19 1 "lhs(%) = factormat(map(collect,rhs(%),[r, v[r], h, c
os(i)],simplify));" "6#/-%$lhsG6#%\"%G-%*factormatG6#-%$mapG6&%(collec
tG-%$rhsG6#F'7&%\"rG&%\"vG6#F3%\"hG-%$cosG6#%\"iG%)simplifyG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"hG6#%\"xG7#&F*6#%\"yG7#
&F*6#%\"zG*&F*\"\"\"-F%6#7%7#*&-%$sinG6#%&OmegaGF6-F=6#%\"iGF67#,$*&F@
\"\"\"-%$cosGF>F6!\"\"7#-FHFAF6" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
19 1 "hvec_elmnts := %:" "6#>%,hvec_elmntsG%\"%G" }}}}{SECT 0 {PARA 4 
"" 0 "2.5. vis viva integral" {TEXT -1 22 "2.5. vis viva integral" }}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "mat2eqs(vvec);" "6#-%(mat2eqsG6#
%%vvecG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%/&%\"vG6#%\"xG,&*&*&,&*&-
%$cosG6#%&OmegaG\"\"\"-%$sinG6#,&%&omegaGF2%&thetaGF2F2!\"\"*(-F4F0F2-
F/F5F2-F/6#%\"iGF2F9F2%\"hGF2\"\"\"%\"rG!\"\"F2*&&F&6#FBF2,&*&F.FAF<FA
F2*(F;FAF3FAF=FAF9F2F2/&F&6#%\"yG,&*&*&,&*&F;FAF3FAF9*(F.FAF<FAF=FAF2F
2F@FAFAFBFCF2*&FEFA,&*&F;FAF<FAF2*(F.FAF3FAF=FAF2F2F2/&F&6#%\"zG*&,&*&
FEFAF3FAF2*&*&F<FAF@FAFAFBFCF2F2-F4F>F2" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "veqs := %:" "6#>%%veqsG%\"%G" }}}{EXCHG {PARA 0 "" 0 "
" {XPPEDIT 19 1 "v^2 = add(rhs(veqs[k])^2,k = 1 .. 3);" "6#/*$%\"vG\"
\"#-%$addG6$*$-%$rhsG6#&%%veqsG6#%\"kG\"\"#/F1;\"\"\"\"\"$" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#/*$)%\"vG\"\"#\"\"\",(*$),&*&*&,&*&-%$cosG6#
%&OmegaG\"\"\"-%$sinG6#,&%&omegaGF5%&thetaGF5F5!\"\"*(-F7F3F5-F2F8F5-F
26#%\"iGF5F<F5%\"hGF5F(%\"rG!\"\"F5*&&F&6#FDF5,&*&F1F(F?F(F5*(F>F(F6F(
F@F(F<F5F5F'F(F5*$),&*&*&,&*&F>F(F6F(F<*(F1F(F?F(F@F(F5F5FCF(F(FDFEF5*
&FGF(,&*&F>F(F?F(F5*(F1F(F6F(F@F(F5F5F5F'F(F5*&),&*&FGF(F6F(F5*&*&F?F(
FCF(F(FDFEF5F'F()-F7FAF'F(F5" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 
"collect(%,[v[r]],simplify);" "6#-%(collectG6%%\"%G7#&%\"vG6#%\"rG%)si
mplifyG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"vG\"\"#\"\"\",&*$)&F
&6#%\"rGF'F(\"\"\"*&*$)%\"hGF'F(F(*$)F.\"\"#F(!\"\"F/" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "v2eq := %:" "6#>%%v2eqG%\"%G" }}}{SECT 
0 {PARA 5 "" 0 "" {TEXT -1 17 "elliptical motion" }}{EXCHG {PARA 0 "" 
0 "" {XPPEDIT 19 1 "simplify(subs(vresub,resub,hesub,v2eq));" "6#-%)si
mplifyG6#-%%subsG6&%'vresubG%&resubG%&hesubG%%v2eqG" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/*$)%\"vG\"\"#\"\"\",$*&*&%'lambdaG\"\"\",(*$)%\"eGF'
F(F-F-F-*&F1F--%$cosG6#%&thetaGF-F'F-F(*&%\"aG\"\"\",&!\"\"F-F/F-\"\"
\"!\"\"F;" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(isolate(resub
,cos(theta)),%);" "6#-%%subsG6$-%(isolateG6$%&resubG-%$cosG6#%&thetaG%
\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"vG\"\"#\"\"\",$*&*&%'la
mbdaG\"\"\",(*$)%\"eGF'F(F-!\"\"F-*&*&%\"aGF-,&F-F-F/F2F-F(%\"rG!\"\"F
'F-F(*&F5\"\"\",&F2F-F/F-\"\"\"F8F2" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "collect(%,[lambda, r],simplify);" "6#-%(collectG6%%\"%G
7$%'lambdaG%\"rG%)simplifyG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"
vG\"\"#\"\"\"*&,&*&F(F(%\"aG!\"\"!\"\"*&F(F(%\"rGF-F'\"\"\"%'lambdaGF1
" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "v2esub := %:" "6#>%'v2esubG
%\"%G" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 17 "hyperbolic motion" }}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "simplify(subs(vrhsub, rhsub, hhs
ub, v2eq));" "6#-%)simplifyG6#-%%subsG6&%'vrhsubG%&rhsubG%&hhsubG%%v2e
qG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"vG\"\"#\"\"\"*&*&%'lambda
G\"\"\",(*$)%\"eGF'F(F,F,F,*&F0F,-%$cosG6#%&thetaGF,F'F,F(*&%\"aG\"\"
\",&!\"\"F,F.F,\"\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "s
ubs(isolate(rhsub,cos(theta)),%);" "6#-%%subsG6$-%(isolateG6$%&rhsubG-
%$cosG6#%&thetaG%\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"vG\"\"
#\"\"\"*&*&%'lambdaG\"\"\",(*$)%\"eGF'F(F,!\"\"F,*&*&%\"aGF,,&F1F,F.F,
F,F(%\"rG!\"\"F'F,F(*&F4\"\"\"F5\"\"\"F7" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "collect(%,[lambda, r],simplify);" "6#-%(collectG6%%\"%G
7$%'lambdaG%\"rG%)simplifyG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"
vG\"\"#\"\"\"*&,&*&F(F(%\"aG!\"\"\"\"\"*&F(F(%\"rGF-F'F.%'lambdaGF." }
}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "v2hsub := %:" "6#>%'v2hsubG%\"
%G" }}}}}{SECT 0 {PARA 4 "" 0 "2.6. Laplace vector" {TEXT -1 19 "2.6. \+
Laplace vector" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "The Laplace-Rung
e-Lenz vector (a constant of two-body motion) is " }{XPPEDIT 18 0 "A =
 cross(v,h)-lambda*r/mag(r);" "6#/%\"AG,&-%&crossG6$%\"vG%\"hG\"\"\"*(
%'lambdaGF+%\"rGF+-%$magG6#F.!\"\"F2" }{TEXT -1 10 ".  Recall " }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "rvec;" "6#%%rvecG" }}{PARA 0 "" 
0 "" {XPPEDIT 19 1 "hvec_elmnts;" "6#%,hvec_elmntsG" }}{PARA 0 "" 0 "
" {XPPEDIT 19 1 "vvec;" "6#%%vvecG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/-%'matrixG6#7%7#%\"xG7#%\"yG7#%\"zG*&%\"rG\"\"\"-F%6#7%7#,&*&-%$cosG6
#%&OmegaGF0-F86#,&%&omegaGF0%&thetaGF0F0F0*(-%$sinGF9F0-FBF<F0-F86#%\"
iGF0!\"\"7#,&*&FA\"\"\"F;FKF0*(F7FKFCFKFDFKF07#*&FCFK-FBFEF0F0" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"hG6#%\"xG7#&F*6#%
\"yG7#&F*6#%\"zG*&F*\"\"\"-F%6#7%7#*&-%$sinG6#%&OmegaGF6-F=6#%\"iGF67#
,$*&F@\"\"\"-%$cosGF>F6!\"\"7#-FHFAF6" }}{PARA 12 "" 1 "" {XPPMATH 20 
"6#/-%'matrixG6#7%7#&%\"vG6#%\"xG7#&F*6#%\"yG7#&F*6#%\"zG-F%6#7%7#,&*&
*&,&*&-%$cosG6#%&OmegaG\"\"\"-%$sinG6#,&%&omegaGFB%&thetaGFBFB!\"\"*(-
FDF@FB-F?FEFB-F?6#%\"iGFBFIFB%\"hGFB\"\"\"%\"rG!\"\"FB*&&F*6#FRFB,&*&F
>FQFLFQFB*(FKFQFCFQFMFQFIFBFB7#,&*&*&,&*&FKFQFCFQFI*(F>FQFLFQFMFQFBFBF
PFQFQFRFSFB*&FUFQ,&*&FKFQFLFQFB*(F>FQFCFQFMFQFBFBFB7#*&,&*&FUFQFCFQFB*
&*&FLFQFPFQFQFRFSFBFB-FDFNFB" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "He
nce," }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "matrix([[A[x]], [A[y]],
 [A[z]]]) = cross(convert(evalm(rhs(vvec)),vector),convert(evalm(rhs(h
vec_elmnts)),vector))-lambda*convert(rhs(rvec)/r,vector):" "6#/-%'matr
ixG6#7%7#&%\"AG6#%\"xG7#&F*6#%\"yG7#&F*6#%\"zG,&-%&crossG6$-%(convertG
6$-%&evalmG6#-%$rhsG6#%%vvecG%'vectorG-F:6$-F=6#-F@6#%,hvec_elmntsGFC
\"\"\"*&%'lambdaGFK-F:6$*&-F@6#%%rvecGFK%\"rG!\"\"FCFKFU" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "Collect(evalm(%),[lambda, h, v[r]],simp
lify):" "6#-%(CollectG6%-%&evalmG6#%\"%G7%%'lambdaG%\"hG&%\"vG6#%\"rG%
)simplifyG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "lhs(%) = mat(rhs(
%));" "6#/-%$lhsG6#%\"%G-%$matG6#-%$rhsG6#F'" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"AG6#%\"xG7#&F*6#%\"yG7#&F*6#%\"zG-
F%6#7%7#,(*&,&*&-%$cosG6#%&OmegaG\"\"\"-F>6#,&%&omegaGFA%&thetaGFAFA!
\"\"*(-%$sinGF?FA-FJFCFA-F>6#%\"iGFAFAFA%'lambdaGFAFA*&*&,&F<FAFHFGFA)
%\"hG\"\"#\"\"\"FV%\"rG!\"\"FA*(,&*(FIFVFBFVFLFVFA*&F=FVFKFVFAFA&%\"vG
6#FWFAFTFAFA7#,(*&,&*&FIFVFBFVFG*(F=FVFKFVFLFVFGFAFOFVFA*&*&,&F^oFAF_o
FAFAFSFVFVFWFXFA*(,&*&FIFVFKFVFA*(F=FVFBFVFLFVFGFAFgnFVFTFVFA7#,(*(FOF
VFKFV-FJFMFAFG*&*(FKFVFjoFVFSFVFVFWFXFA**FjoFVFBFVFgnFVFTFVFG" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Collect(subs(vresub,resub,hesub,
%),lambda);" "6#-%(CollectG6$-%%subsG6&%'vresubG%&resubG%&hesubG%\"%G%
'lambdaG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"AG6#%
\"xG7#&F*6#%\"yG7#&F*6#%\"zG-F%6#7%7#*&,**&-%$cosG6#%&OmegaG\"\"\"-F=6
#,&%&omegaGF@%&thetaGF@F@!\"\"*(-%$sinGF>F@-FIFBF@-F=6#%\"iGF@F@*&,&F;
F@FGFFF@,&F@F@*&%\"eGF@-F=6#FEF@F@F@F@*(,&*(FH\"\"\"FAFXFKFXF@*&F<FXFJ
FXF@F@FRFX-FIFTF@F@F@%'lambdaGF@7#*&,**&FHFXFAFXFF*(F<FXFJFXFKFXFF*&,&
FinF@FjnF@F@FPFXF@*(,&*&FHFXFJFXF@*(F<FXFAFXFKFXFFF@FRFXFZFXF@F@FenFX7
#*&,(*&FJFX-FIFLF@FF*(FJFXFeoFXFPFXF@**FeoFXFAFXFRFXFZFXFFF@FenFX" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "factormat(Collect(%,[lambda, e, \+
sin(theta), cos(theta)],factor));" "6#-%*factormatG6#-%(CollectG6%%\"%
G7&%'lambdaG%\"eG-%$sinG6#%&thetaG-%$cosG6#F0%'factorG" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"AG6#%\"xG7#&F*6#%\"yG7#&F*6#%
\"zG*(%'lambdaG\"\"\"%\"eGF7-F%6#7%7#,&*&,&*&-%$cosG6#%&OmegaGF7-FB6#,
&%&omegaGF7%&thetaGF7F7F7*(-%$sinGFCF7-FLFFF7-FB6#%\"iGF7!\"\"F7-FB6#F
IF7F7*&,&*(FK\"\"\"FEFWFNFWF7*&FAFWFMFWF7F7-FLFSF7F77#,&*&,&*&FKFWFEFW
F7*(FAFWFMFWFNFWF7F7FRFWF7*&,&*&FKFWFMFWF7*(FAFWFEFWFNFWFQF7FYFWF77#,&
*(FMFW-FLFOF7FRFWF7*(FaoFWFEFWFYFWFQF7" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "Avec_elmnts := %:" "6#>%,Avec_elmntsG%\"%G" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 36 "The magnitude of the Laplace vector:" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "convert(evalm(rhs(Avec_elmnts)),
vector):" "6#-%(convertG6$-%&evalmG6#-%$rhsG6#%,Avec_elmntsG%'vectorG
" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "A = simplify(sqrt(simplify(
dot(%,%))),assume = positive);" "6#/%\"AG-%)simplifyG6$-%%sqrtG6#-F&6#
-%$dotG6$%\"%GF0/%'assumeG%)positiveG" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#/%\"AG*&%'lambdaG\"\"\"%\"eGF'" }}}}}{SECT 0 {PARA 3 "" 0 "3. Solve
 for the Orbital Elements" {TEXT -1 33 "3. Solve for the Orbital Eleme
nts" }}{SECT 0 {PARA 4 "" 0 "3.1. summary of useful equations" {TEXT 
-1 32 "3.1. summary of useful equations" }}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 1 0 110 "hvec; hvec_elmnts; rvec; vvec; hesub; hhsub; \nresub
; rhsub; vresub; vrhsub; v2eq; v2esub; v2hsub; Avec_elmnts;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"hG6#%\"xG7#&F*6#%\"yG7#
&F*6#%\"zG-F%6#7%7#,&*&F0\"\"\"&%\"vG6#F4F;F;*&F4F;&F=6#F0F;!\"\"7#,&*
&F4\"\"\"&F=6#F,F;F;*&F,F;F<FFFB7#,&*&F,FFF@FFF;*&F0FFFGFFFB" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"hG6#%\"xG7#&F*6#%\"yG7#
&F*6#%\"zG*&F*\"\"\"-F%6#7%7#*&-%$sinG6#%&OmegaGF6-F=6#%\"iGF67#,$*&F@
\"\"\"-%$cosGF>F6!\"\"7#-FHFAF6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
'matrixG6#7%7#%\"xG7#%\"yG7#%\"zG*&%\"rG\"\"\"-F%6#7%7#,&*&-%$cosG6#%&
OmegaGF0-F86#,&%&omegaGF0%&thetaGF0F0F0*(-%$sinGF9F0-FBF<F0-F86#%\"iGF
0!\"\"7#,&*&FA\"\"\"F;FKF0*(F7FKFCFKFDFKF07#*&FCFK-FBFEF0F0" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"vG6#%\"xG7#&F*6#%\"yG7#
&F*6#%\"zG-F%6#7%7#,&*&*&,&*&-%$cosG6#%&OmegaG\"\"\"-%$sinG6#,&%&omega
GFB%&thetaGFBFB!\"\"*(-FDF@FB-F?FEFB-F?6#%\"iGFBFIFB%\"hGFB\"\"\"%\"rG
!\"\"FB*&&F*6#FRFB,&*&F>FQFLFQFB*(FKFQFCFQFMFQFIFBFB7#,&*&*&,&*&FKFQFC
FQFI*(F>FQFLFQFMFQFBFBFPFQFQFRFSFB*&FUFQ,&*&FKFQFLFQFB*(F>FQFCFQFMFQFB
FBFB7#*&,&*&FUFQFCFQFB*&*&FLFQFPFQFQFRFSFBFB-FDFNFB" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/%\"hG*$-%%sqrtG6#*(%'lambdaG\"\"\"%\"aGF+,&F+F+*$)%
\"eG\"\"#\"\"\"!\"\"F+F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"hG*$-%
%sqrtG6#*(%'lambdaG\"\"\"%\"aGF+,&!\"\"F+*$)%\"eG\"\"#\"\"\"F+F+F3" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"rG*&*&%\"aG\"\"\",&F(F(*$)%\"eG\"
\"#\"\"\"!\"\"F(F.,&F(F(*&F,F(-%$cosG6#%&thetaGF(F(!\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/%\"rG*&*&%\"aG\"\"\",&!\"\"F(*$)%\"eG\"\"#\"\"
\"F(F(F/,&F(F(*&F-F(-%$cosG6#%&thetaGF(F(!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%\"vG6#%\"rG*&*(%\"eG\"\"\"-%$sinG6#%&thetaGF+-%%sqrt
G6#*(%'lambdaGF+%\"aGF+,&F+F+*$)F*\"\"#\"\"\"!\"\"F+F:F:*&F5\"\"\"F6\"
\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"vG6#%\"rG*&*(%\"eG\"
\"\"-%$sinG6#%&thetaGF+-%%sqrtG6#*(%'lambdaGF+%\"aGF+,&!\"\"F+*$)F*\"
\"#\"\"\"F+F+F;F;*&F5\"\"\"F6\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/*$)%\"vG\"\"#\"\"\",&*$)&F&6#%\"rGF'F(\"\"\"*&*$)%\"hGF'F(F(*$)
F.\"\"#F(!\"\"F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%\"vG\"\"#\"\"
\"*&,&*&F(F(%\"aG!\"\"!\"\"*&F(F(%\"rGF-F'\"\"\"%'lambdaGF1" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*$)%\"vG\"\"#\"\"\"*&,&*&F(F(%\"aG!\"\"\"\"
\"*&F(F(%\"rGF-F'F.%'lambdaGF." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%'
matrixG6#7%7#&%\"AG6#%\"xG7#&F*6#%\"yG7#&F*6#%\"zG*(%'lambdaG\"\"\"%\"
eGF7-F%6#7%7#,&*&,&*&-%$cosG6#%&OmegaGF7-FB6#,&%&omegaGF7%&thetaGF7F7F
7*(-%$sinGFCF7-FLFFF7-FB6#%\"iGF7!\"\"F7-FB6#FIF7F7*&,&*(FK\"\"\"FEFWF
NFWF7*&FAFWFMFWF7F7-FLFSF7F77#,&*&,&*&FKFWFEFWF7*(FAFWFMFWFNFWF7F7FRFW
F7*&,&*&FKFWFMFWF7*(FAFWFEFWFNFWFQF7FYFWF77#,&*(FMFW-FLFOF7FRFWF7*(Fao
FWFEFWFYFWFQF7" }}}}{SECT 0 {PARA 4 "" 0 "3.2. eccentricity" {TEXT -1 
17 "3.2. eccentricity" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "The defin
ition of the Laplace vector is " }{XPPEDIT 18 0 "A = cross(v,h)-lambda
*r/mag(r);" "6#/%\"AG,&-%&crossG6$%\"vG%\"hG\"\"\"*(%'lambdaGF+%\"rGF+
-%$magG6#F.!\"\"F2" }{TEXT -1 28 ".  Hence, in component form," }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "matrix([[A[x]], [A[y]], [A[z]]])
 = mat(cross([v[x], v[y], v[z]],[h[x], h[y], h[z]]))-lambda*matrix([[x
], [y], [z]])/sqrt(x^2+y^2+z^2):" "6#/-%'matrixG6#7%7#&%\"AG6#%\"xG7#&
F*6#%\"yG7#&F*6#%\"zG,&-%$matG6#-%&crossG6$7%&%\"vG6#F,&F>6#F0&F>6#F47
%&%\"hG6#F,&FF6#F0&FF6#F4\"\"\"*(%'lambdaGFL-F%6#7%7#F,7#F07#F4FL-%%sq
rtG6#,(*$F,\"\"#FL*$F0\"\"#FL*$F4\"\"#FL!\"\"Fin" }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "evalm(%);" "6#-%&evalmG6#%\"%G" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"AG6#%\"xG7#&F*6#%\"yG7#&F*6#%\"
zG-F%6#7%7#,(*&&%\"vG6#F0\"\"\"&%\"hGF3F>F>*&&F<6#F4F>&F@F/F>!\"\"*&*&
%'lambdaGF>F,F>\"\"\"*$-%%sqrtG6#,(*$)F,\"\"#FIF>*$)F0FQFIF>*$)F4FQFIF
>FI!\"\"FE7#,(*&FBFI&F@F+F>F>*&&F<6#F,F>F?FIFE*&*&FHFIF0F>FI*$-FL6#FNF
IFVFE7#,(*&FfnFIFDFIF>*&F;FIFZFIFE*&*&FHFIF4F>FI*$-FL6#FNFIFVFE" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "We also have that" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "hvec;" "6#%%hvecG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"hG6#%\"xG7#&F*6#%\"yG7#&F*6#%\"zG-
F%6#7%7#,&*&F0\"\"\"&%\"vG6#F4F;F;*&F4F;&F=6#F0F;!\"\"7#,&*&F4\"\"\"&F
=6#F,F;F;*&F,F;F<FFFB7#,&*&F,FFF@FFF;*&F0FFFGFFFB" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 10 "Therefore," }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 
1 "subs(mat2eqs(%),%%);" "6#-%%subsG6$-%(mat2eqsG6#%\"%G%#%%G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"AG6#%\"xG7#&F*6#%
\"yG7#&F*6#%\"zG-F%6#7%7#,(*&&%\"vG6#F0\"\"\",&*&F,\"\"\"F;FAFA*&F0FA&
F<6#F,FA!\"\"FAFA*&&F<6#F4F>,&*&F4FAFCF>FA*&F,F>FGFAFEFAFE*&*&%'lambda
GFAF,F>F>*$-%%sqrtG6#,(*$)F,\"\"#F>FA*$)F0FVF>FA*$)F4FVF>FAF>!\"\"FE7#
,(*&FGF>,&*&F0F>FGF>FA*&F4F>F;F>FEFAFA*&FCF>F?F>FE*&*&FNF>F0F>F>*$-FQ6
#FSF>FenFE7#,(*&FCF>FIF>FA*&F;F>FinF>FE*&*&FNF>F4F>F>*$-FQ6#FSF>FenFE
" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Avec := %:" "6#>%%AvecG%\"%
G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Since the magnitude of this \+
vector is " }{XPPEDIT 18 0 "A = lambda*e;" "6#/%\"AG*&%'lambdaG\"\"\"%
\"eGF'" }{TEXT -1 41 ", we may define an \"eccentricity vector\":" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "matrix([[e[x]], [e[y]], [e[z]]])
 = rhs(Avec)/lambda;" "6#/-%'matrixG6#7%7#&%\"eG6#%\"xG7#&F*6#%\"yG7#&
F*6#%\"zG*&-%$rhsG6#%%AvecG\"\"\"%'lambdaG!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"eG6#%\"xG7#&F*6#%\"yG7#&F*6#%\"zG*
&-F%6#7%7#,(*&&%\"vG6#F0\"\"\",&*&F,\"\"\"F<FBFB*&F0FB&F=6#F,FB!\"\"FB
FB*&&F=6#F4F?,&*&F4FBFDF?FB*&F,F?FHFBFFFBFF*&*&%'lambdaGFBF,F?F?*$-%%s
qrtG6#,(*$)F,\"\"#F?FB*$)F0FWF?FB*$)F4FWF?FBF?!\"\"FF7#,(*&FHF?,&*&F0F
?FHF?FB*&F4F?F<F?FFFBFB*&FDF?F@F?FF*&*&FOF?F0F?F?*$-FR6#FTF?FfnFF7#,(*
&FDF?FJF?FB*&F<F?FjnF?FF*&*&FOF?F4F?F?*$-FR6#FTF?FfnFFF?FOFfn" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "from which we obtain the magnitude
 of the eccentricity, " }{XPPEDIT 18 0 "e = sqrt(e[x]^2+e[y]^2+e[z]^2)
;" "6#/%\"eG-%%sqrtG6#,(*$&F$6#%\"xG\"\"#\"\"\"*$&F$6#%\"yG\"\"#F.*$&F
$6#%\"zG\"\"#F." }{TEXT -1 3 ".  " }}}{SECT 0 {PARA 5 "" 0 "" {TEXT 
-1 28 "An attempt at simplification" }}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "convert(evalm(rhs(%)),vector):" "6#-%(convertG6$-%&eval
mG6#-%$rhsG6#%\"%G%'vectorG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "%[1]^2+%
[2]^2+%[3]^2:" "6#,(*$&%\"%G6#\"\"\"\"\"#\"\"\"*$&F&6#\"\"#\"\"#F**$&F
&6#\"\"$\"\"#F*" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "cost(%);" "6
#-%%costG6#%\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%*additionsG\"#?
%*divisionsG\"\"$%0multiplicationsG\"#U%*functionsG\"#m" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "Collect(%%,[1/sqrt(x^2+y^2+z^2), lambda
],factor,loc);" "6#-%(CollectG6&%#%%G7$*&\"\"\"\"\"\"-%%sqrtG6#,(*$%\"
xG\"\"#F**$%\"yG\"\"#F**$%\"zG\"\"#F*!\"\"%'lambdaG%'factorG%$locG" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#,(\"\"\"F$*&,4*&)%\"xG\"\"#\"\"\")&%\"
vG6#%\"yGF*F+!\"#**F)F$F-F$F0F$&F.6#F)F$\"\"%**F)F+&F.6#%\"zGF$F9F$F3F
+F5*&F(F+)F7F*F+F1*&)F0F*F+F;F+F1**F7F+F9F+F-F+F0F+F5*&F=F+)F3F*F+F1*&
)F9F*F+F@F+F1*&FBF+F,F+F1F+*&%'lambdaG\"\"\"-%%sqrtG6#,(*$F(F+F$*$F=F+
F$*$FBF+F$F+!\"\"F$*&*&,(*$F@F+F$*$F,F+F$*$F;F+F$F$,4F'F$F2F1F6F1F:F$F
<F$F>F1F?F$FAF$FCF$F$F+*$)FE\"\"#F+FNF$" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "x^2*v[y]^2-2*x*v[y]*y*v[x]-2*x*v[z]*z*v[x]+x^2*v[z]^2+y
^2*v[z]^2-2*v[z]*z*v[y]*y+y^2*v[x]^2+z^2*v[x]^2+z^2*v[y]^2:" "6#,4*&%
\"xG\"\"#&%\"vG6#%\"yG\"\"#\"\"\"*,\"\"#F,F%F,&F(6#F*F,F*F,&F(6#F%F,!
\"\"*,\"\"#F,F%F,&F(6#%\"zGF,F8F,&F(6#F%F,F3*&F%\"\"#&F(6#F8\"\"#F,*&F
*\"\"#&F(6#F8\"\"#F,*,\"\"#F,&F(6#F8F,F8F,&F(6#F*F,F*F,F3*&F*\"\"#&F(6
#F%\"\"#F,*&F8\"\"#&F(6#F%\"\"#F,*&F8\"\"#&F(6#F*\"\"#F," }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "csquare(%,[v[x]^2+v[y]^2+v[z]^2, x^2+y^
2+z^2],factor);" "6#-%(csquareG6%%\"%G7$,(*$&%\"vG6#%\"xG\"\"#\"\"\"*$
&F+6#%\"yG\"\"#F/*$&F+6#%\"zG\"\"#F/,(*$F-\"\"#F/*$F3\"\"#F/*$F8\"\"#F
/%'factorG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,.*&,&*$)&%\"vG6#%\"zG\"
\"#\"\"\"\"\"\"*$)&F)6#%\"yGF,F-F.F.,(*$)%\"xGF,F-F.*$)F3F,F-F.*$)F+F,
F-F.F.F.*&,&F&!\"\"*$)&F)6#F7F,F-F.F.F;F-F.*&,&F?F.F/F>F.F9F-F.**F(F.F
+F.F1F.F3F.!\"#**F7F.F(F-F+F-FAF.FF**F7F-F1F-F3F-FAF-FF" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(%% = %,%%%);" "6#-%%subsG6$/%#%%G%
\"%G%$%%%G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(\"\"\"F$*&,4*&)%\"xG\"
\"#\"\"\")&%\"vG6#%\"yGF*F+!\"#**F)F$F-F$F0F$&F.6#F)F$\"\"%**F)F+&F.6#
%\"zGF$F9F$F3F+F5*&F(F+)F7F*F+F1*&)F0F*F+F;F+F1**F7F+F9F+F-F+F0F+F5*&F
=F+)F3F*F+F1*&)F9F*F+F@F+F1*&FBF+F,F+F1F+*&%'lambdaG\"\"\"-%%sqrtG6#,(
*$F(F+F$*$F=F+F$*$FBF+F$F+!\"\"F$*&*&,(*$F@F+F$*$F,F+F$*$F;F+F$F$,.*&,
&FTF$FSF$F$FJF$F$*&,&FT!\"\"FRF$F$FBF+F$*&,&FRF$FSFZF$F=F+F$F>F1F6F1F2
F1F$F+*$)FE\"\"#F+FNF$" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "-2*x^
2*v[y]^2+4*x*v[y]*y*v[x]+4*x*v[z]*z*v[x]-2*x^2*v[z]^2-2*y^2*v[z]^2+4*v
[z]*z*v[y]*y-2*y^2*v[x]^2-2*z^2*v[x]^2-2*z^2*v[y]^2:" "6#,4*(\"\"#\"\"
\"*$%\"xG\"\"#F&&%\"vG6#%\"yG\"\"#!\"\"*,\"\"%F&F(F&&F+6#F-F&F-F&&F+6#
F(F&F&*,\"\"%F&F(F&&F+6#%\"zGF&F:F&&F+6#F(F&F&*(\"\"#F&*$F(\"\"#F&&F+6
#F:\"\"#F/*(\"\"#F&*$F-\"\"#F&&F+6#F:\"\"#F/*,\"\"%F&&F+6#F:F&F:F&&F+6
#F-F&F-F&F&*(\"\"#F&*$F-\"\"#F&&F+6#F(\"\"#F/*(\"\"#F&*$F:\"\"#F&&F+6#
F(\"\"#F/*(\"\"#F&*$F:\"\"#F&&F+6#F-\"\"#F/" }}}{EXCHG {PARA 0 "" 0 "
" {XPPEDIT 19 1 "csquare(%,[v[x]^2+v[y]^2+v[z]^2, x^2+y^2+z^2],factor)
;" "6#-%(csquareG6%%\"%G7$,(*$&%\"vG6#%\"xG\"\"#\"\"\"*$&F+6#%\"yG\"\"
#F/*$&F+6#%\"zG\"\"#F/,(*$F-\"\"#F/*$F3\"\"#F/*$F8\"\"#F/%'factorG" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#,.*&,&*$)&%\"vG6#%\"zG\"\"#\"\"\"!\"#*
$)&F)6#%\"yGF,F-F.\"\"\",(*$)%\"xGF,F-F4*$)F3F,F-F4*$)F+F,F-F4F4F4*&,&
F&F,*$)&F)6#F8F,F-F.F4F<F-F4*&,&F?F.F/F,F4F:F-F4**F(F4F+F4F1F4F3F4\"\"
%**F8F4F(F-F+F-FAF4FF**F8F-F1F-F3F-FAF-FF" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "subs(%% = %,%%%);" "6#-%%subsG6$/%#%%G%\"%G%$%%%G" }}
{PARA 12 "" 1 "" {XPPMATH 20 "6#,(\"\"\"F$*&,.*&,&*$)&%\"vG6#%\"zG\"\"
#\"\"\"!\"#*$)&F,6#%\"yGF/F0F1F$,(*$)%\"xGF/F0F$*$)F6F/F0F$*$)F.F/F0F$
F$F$*&,&F)F/*$)&F,6#F:F/F0F1F$F>F0F$*&,&FAF1F2F/F$F<F0F$**F+F$F.F$F4F$
F6F$\"\"%**F:F$F+F0F.F0FCF$FH**F:F0F4F0F6F0FCF0FHF0*&%'lambdaG\"\"\"-%
%sqrtG6#F7F0!\"\"F$*&*&,(FAF$F2F$F)F$F$,.*&,&F)F$F2F$F$F7F0F$*&,&F)!\"
\"FAF$F$F>F0F$*&,&FAF$F2FZF$F<F0F$FGF1FIF1FJF1F$F0*$)FL\"\"#F0FQF$" }}
}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Collect(%,[lambda, v[x]^2+v[y]^
2+v[z]^2, x^2+y^2+z^2],loc);" "6#-%(CollectG6%%\"%G7%%'lambdaG,(*$&%\"
vG6#%\"xG\"\"#\"\"\"*$&F,6#%\"yG\"\"#F0*$&F,6#%\"zG\"\"#F0,(*$F.\"\"#F
0*$F4\"\"#F0*$F9\"\"#F0%$locG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,(\"
\"\"F$*&,&*&,&*$)&%\"vG6#%\"zG\"\"#\"\"\"!\"#*$)&F,6#%\"yGF/F0F1F$-%%s
qrtG6#,(*$)%\"xGF/F0F$*$)F6F/F0F$*$)F.F/F0F$F0F$*&,,*&,&F)F/*$)&F,6#F=
F/F0F1F$FAF0F$*&,&FFF1F2F/F$F?F0F$**F+F$F.F$F4F$F6F$\"\"%**F=F$F+F0F.F
0FHF$FM**F=F0F4F0F6F0FHF0FMF0*$-F86#F:F0!\"\"F$F0%'lambdaGFSF$*&*&,(FF
F$F2F$F)F$F$,.*&,&F)F$F2F$F$F:F$F$*&,&F)!\"\"FFF$F$FAF0F$*&,&FFF$F2Fgn
F$F?F0F$FLF1FNF1FOF1F$F0*$)FT\"\"#F0FSF$" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "cost(%);" "6#-%%costG6#%\"%G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,*%*additionsG\"#E%0multiplicationsG\"#p%*functionsG\"#
$)%*divisionsG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "Oh well." }
}}}}{SECT 0 {PARA 4 "" 0 "3.3. semimajor axis" {TEXT -1 19 "3.3. semim
ajor axis" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "elliptical motion:" }
}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "isolate(v2esub,1/a);" "6#-%(is
olateG6$%'v2esubG*&\"\"\"\"\"\"%\"aG!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&\"\"\"F%%\"aG!\"\",&*&*$)%\"vG\"\"#F%F%%'lambdaGF'!
\"\"*&F%F%%\"rGF'F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "hyperbolic
 motion:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "isolate(v2hsub,1/a)
;" "6#-%(isolateG6$%'v2hsubG*&\"\"\"\"\"\"%\"aG!\"\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/*&\"\"\"F%%\"aG!\"\",&*&*$)%\"vG\"\"#F%F%%'lambdaGF
'\"\"\"*&F%F%%\"rGF'!\"#" }}}}{SECT 0 {PARA 4 "" 0 "3.4. inclination" 
{TEXT -1 16 "3.4. inclination" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "W
e can determine the orbital inclination from the angular momentum vect
or, calculated from" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "hvec;" "
6#%%hvecG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"hG6#
%\"xG7#&F*6#%\"yG7#&F*6#%\"zG-F%6#7%7#,&*&F0\"\"\"&%\"vG6#F4F;F;*&F4F;
&F=6#F0F;!\"\"7#,&*&F4\"\"\"&F=6#F,F;F;*&F,F;F<FFFB7#,&*&F,FFF@FFF;*&F
0FFFGFFFB" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Since we know that" 
}}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "hvec_elmnts;" "6#%,hvec_elmnt
sG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#&%\"hG6#%\"xG7#
&F*6#%\"yG7#&F*6#%\"zG*&F*\"\"\"-F%6#7%7#*&-%$sinG6#%&OmegaGF6-F=6#%\"
iGF67#,$*&F@\"\"\"-%$cosGF>F6!\"\"7#-FHFAF6" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 45 "and since the inclination is in the range [0," }
{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 20 "], we therefore have" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "isolate(subs(mat2eqs(hvec),mat2e
qs(hvec_elmnts)[3]),cos(i));" "6#-%(isolateG6$-%%subsG6$-%(mat2eqsG6#%
%hvecG&-F*6#%,hvec_elmntsG6#\"\"$-%$cosG6#%\"iG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$cosG6#%\"iG,$*&,&*&%\"xG\"\"\"&%\"vG6#%\"yGF-!\"\"*
&F1F-&F/6#F,F-F-\"\"\"%\"hG!\"\"F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 22 "For elliptical motion," }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 
1 "subs(hesub,%);" "6#-%%subsG6$%&hesubG%\"%G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$cosG6#%\"iG,$*&,&*&%\"xG\"\"\"&%\"vG6#%\"yGF-!\"\"*
&F1F-&F/6#F,F-F-\"\"\"*$-%%sqrtG6#*(%'lambdaGF-%\"aGF-,&F-F-*$)%\"eG\"
\"#F6F2F-F6!\"\"F2" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "For hyperbo
lic motion," }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(hhsub,%%);
" "6#-%%subsG6$%&hhsubG%#%%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$co
sG6#%\"iG,$*&,&*&%\"xG\"\"\"&%\"vG6#%\"yGF-!\"\"*&F1F-&F/6#F,F-F-\"\"
\"*$-%%sqrtG6#*(%'lambdaGF-%\"aGF-,&F2F-*$)%\"eG\"\"#F6F-F-F6!\"\"F2" 
}}}}{SECT 0 {PARA 4 "" 0 "3.5. ascending node" {TEXT -1 19 "3.5. ascen
ding node" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "mat2eqs(hvec_elmnts
);" "6#-%(mat2eqsG6#%,hvec_elmntsG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
7%/&%\"hG6#%\"xG*(F&\"\"\"-%$sinG6#%&OmegaGF*-F,6#%\"iGF*/&F&6#%\"yG,$
*(F&\"\"\"F/F8-%$cosGF-F*!\"\"/&F&6#%\"zG*&F&F8-F:F0F*" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "solve(\{%[1], %[2]\},\{sin(Omega), cos(
Omega)\});" "6#-%&solveG6$<$&%\"%G6#\"\"\"&F(6#\"\"#<$-%$sinG6#%&Omega
G-%$cosG6#F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%$cosG6#%&OmegaG,$
*&&%\"hG6#%\"yG\"\"\"*&F,\"\"\"-%$sinG6#%\"iG\"\"\"!\"\"!\"\"/-F3F'*&&
F,6#%\"xGF/*&F,\"\"\"F2\"\"\"F7" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 
19 1 "subs(mat2eqs(hvec),%);" "6#-%%subsG6$-%(mat2eqsG6#%%hvecG%\"%G" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%$cosG6#%&OmegaG,$*&,&*&%\"zG\"
\"\"&%\"vG6#%\"xGF.F.*&F2F.&F06#F-F.!\"\"\"\"\"*&%\"hG\"\"\"-%$sinG6#%
\"iG\"\"\"!\"\"F6/-F<F'*&,&*&%\"yGF.F4F7F.*&F-F7&F06#FFF.F6F7*&F9\"\"
\"F;\"\"\"F@" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Elliptical motion
:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(hesub,%);" "6#-%%subs
G6$%&hesubG%\"%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%$sinG6#%&Omeg
aG*&,&*&%\"yG\"\"\"&%\"vG6#%\"zGF-F-*&F1F-&F/6#F,F-!\"\"\"\"\"*&-%%sqr
tG6#*(%'lambdaGF-%\"aGF-,&F-F-*$)%\"eG\"\"#F6F5F-F6-F&6#%\"iG\"\"\"!\"
\"/-%$cosGF',$*&,&*&F1F6&F/6#%\"xGF-F-*&FQF-F.F6F5F6*&-F96#F;F6FC\"\"
\"FGF5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Hyperbolic motion:" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(hhsub,%%);" "6#-%%subsG6$%&
hhsubG%#%%G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/-%$cosG6#%&OmegaG,$*
&,&*&%\"zG\"\"\"&%\"vG6#%\"xGF.F.*&F2F.&F06#F-F.!\"\"\"\"\"*&-%%sqrtG6
#*(%'lambdaGF.%\"aGF.,&F6F.*$)%\"eG\"\"#F7F.F.F7-%$sinG6#%\"iG\"\"\"!
\"\"F6/-FEF'*&,&*&%\"yGF.F4F7F.*&F-F7&F06#FOF.F6F7*&-F:6#F<F7FD\"\"\"F
I" }}}}{SECT 0 {PARA 4 "" 0 "3.6. true anomaly" {TEXT -1 17 "3.6. true
 anomaly" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 15 "circular motion" }}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "rvec;" "6#%%rvecG" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#%\"xG7#%\"yG7#%\"zG*&%\"rG\"\"
\"-F%6#7%7#,&*&-%$cosG6#%&OmegaGF0-F86#,&%&omegaGF0%&thetaGF0F0F0*(-%$
sinGF9F0-FBF<F0-F86#%\"iGF0!\"\"7#,&*&FA\"\"\"F;FKF0*(F7FKFCFKFDFKF07#
*&FCFK-FBFEF0F0" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "subs(omega =
 0,reqs);" "6#-%%subsG6$/%&omegaG\"\"!%%reqsG" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#7%/%\"xG*&%\"rG\"\"\",&*&-%$cosG6#%&OmegaGF(-F,6#%&thet
aGF(F(*(-%$sinGF-F(-F4F0F(-F,6#%\"iGF(!\"\"F(/%\"yG*&F'\"\"\",&*&F3F=F
/F=F(*(F+F=F5F=F6F=F(F(/%\"zG*(F'F=F5F=-F4F7F(" }}}{EXCHG {PARA 0 "" 
0 "" {XPPEDIT 19 1 "isolate(simplify(%[1]*cos(Omega)+%[2]*sin(Omega)),
cos(theta));" "6#-%(isolateG6$-%)simplifyG6#,&*&&%\"%G6#\"\"\"\"\"\"-%
$cosG6#%&OmegaGF/F/*&&F,6#\"\"#F/-%$sinG6#F3F/F/-F16#%&thetaG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%&thetaG,$*&,&*&-F%6#%&Omega
G\"\"\"%\"xGF/!\"\"*&-%$sinGF-F/%\"yGF/F1\"\"\"%\"rG!\"\"F1" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "(%%[2]*cos(Omega)-%%[1]*sin(Omeg
a))*cos(i)+%%[3]*sin(i);" "6#,&*&,&*&&%#%%G6#\"\"#\"\"\"-%$cosG6#%&Ome
gaGF+F+*&&F(6#\"\"\"F+-%$sinG6#F/F+!\"\"F+-F-6#%\"iGF+F+*&&F(6#\"\"$F+
-F56#F:F+F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/,&*&-%$cosG6#%\"iG\"\"
\",&*&-F'6#%&OmegaGF*%\"yGF*F**&-%$sinGF.F*%\"xGF*!\"\"F*F**&-F3F(F*%
\"zGF*F*,&*&F&\"\"\",&*(F-F;%\"rGF*,&*&F2F;-F'6#%&thetaGF*F**(F-F;-F3F
BF*F&F;F*F*F**(F2F;F>F;,&*&F-F;FAF;F**(F2F;FEF;F&F;F5F*F5F*F**()F7\"\"
#F;F>F;FEF;F*" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "collect(isolat
e(%,sin(theta)),[r, cos(i)],simplify);" "6#-%(collectG6%-%(isolateG6$%
\"%G-%$sinG6#%&thetaG7$%\"rG-%$cosG6#%\"iG%)simplifyG" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%$sinG6#%&thetaG*&,&*&,&*&-%$cosG6#%&OmegaG\"\"
\"%\"yGF1!\"\"*&-F%F/F1%\"xGF1F1F1-F.6#%\"iGF1F3*&-F%F8F1%\"zGF1F1\"\"
\"%\"rG!\"\"" }}}}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 17 "elliptical moti
on" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "resub;" "6#%&resubG" }}
{PARA 0 "" 0 "" {XPPEDIT 19 1 "vresub;" "6#%'vresubG" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/%\"rG*&*&%\"aG\"\"\",&F(F(*$)%\"eG\"\"#\"\"\"!\"\"F
(F.,&F(F(*&F,F(-%$cosG6#%&thetaGF(F(!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/&%\"vG6#%\"rG*&*(%\"eG\"\"\"-%$sinG6#%&thetaGF+-%%sqrt
G6#*(%'lambdaGF+%\"aGF+,&F+F+*$)F*\"\"#\"\"\"!\"\"F+F:F:*&F5\"\"\"F6\"
\"\"!\"\"" }}}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
sin(theta);" "6#-%$sinG6#%&thetaG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "isolate(vresub,sin(theta));" "6#-%(isolateG6$%'v
resubG-%$sinG6#%&thetaG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#%
&thetaG*&*(&%\"vG6#%\"rG\"\"\"%\"aG\"\"\",&F.F.*$)%\"eG\"\"#F0!\"\"F0F
0*&F4\"\"\"-%%sqrtG6#*(%'lambdaGF.F/F.F1F.F0!\"\"" }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "collect(subs(v[r] = dot([x/r, y/r, z/r],[v[x], v
[y], v[z]]),%),r);" "6#-%(collectG6$-%%subsG6$/&%\"vG6#%\"rG-%$dotG6$7
%*&%\"xG\"\"\"F-!\"\"*&%\"yGF4F-F5*&%\"zGF4F-F57%&F+6#F3&F+6#F7&F+6#F9
%\"%GF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#%&thetaG*&*(,(*&%
\"xG\"\"\"&%\"vG6#F,F-F-*&%\"yGF-&F/6#F2F-F-*&%\"zGF-&F/6#F6F-F-F-%\"a
G\"\"\",&F-F-*$)%\"eG\"\"#F:!\"\"F:F:*(F>\"\"\"-%%sqrtG6#*(%'lambdaGF-
F9F-F;F-F:%\"rG\"\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "s
ubs(r = sqrt(x^2+y^2+z^2), %);" "6#-%%subsG6$/%\"rG-%%sqrtG6#,(*$%\"xG
\"\"#\"\"\"*$%\"yG\"\"#F/*$%\"zG\"\"#F/%\"%G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$sinG6#%&thetaG*&*(,(*&%\"xG\"\"\"&%\"vG6#F,F-F-*&%
\"yGF-&F/6#F2F-F-*&%\"zGF-&F/6#F6F-F-F-%\"aG\"\"\",&F-F-*$)%\"eG\"\"#F
:!\"\"F:F:*(F>\"\"\"-%%sqrtG6#*(%'lambdaGF-F9F-F;F-F:-FD6#,(*$)F,F?F:F
-*$)F2F?F:F-*$)F6F?F:F-F:!\"\"" }}}}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "cos(theta);" "6#-%$cosG6#%&thetaG" }{TEXT -1 1 "
 " }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "isolate(resub,cos(theta));
" "6#-%(isolateG6$%&resubG-%$cosG6#%&thetaG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/-%$cosG6#%&thetaG*&,&*&*&%\"aG\"\"\",&F-F-*$)%\"eG\"\"
#\"\"\"!\"\"F-F3%\"rG!\"\"F-F4F-F3F1F6" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "subs(r = sqrt(x^2+y^2+z^2),%);" "6#-%%subsG6$/%\"rG-%%s
qrtG6#,(*$%\"xG\"\"#\"\"\"*$%\"yG\"\"#F/*$%\"zG\"\"#F/%\"%G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%&thetaG*&,&*&*&%\"aG\"\"\",&F-F-*
$)%\"eG\"\"#\"\"\"!\"\"F-F3*$-%%sqrtG6#,(*$)%\"xGF2F3F-*$)%\"yGF2F3F-*
$)%\"zGF2F3F-F3!\"\"F-F4F-F3F1FC" }}}}}{SECT 0 {PARA 5 "" 0 "" {TEXT 
-1 17 "hyperbolic motion" }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "rhsu
b;" "6#%&rhsubG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "vrhsub;" "6#%'vrhsub
G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"rG*&*&%\"aG\"\"\",&!\"\"F(*$)
%\"eG\"\"#\"\"\"F(F(F/,&F(F(*&F-F(-%$cosG6#%&thetaGF(F(!\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/&%\"vG6#%\"rG*&*(%\"eG\"\"\"-%$sinG6#%&thet
aGF+-%%sqrtG6#*(%'lambdaGF+%\"aGF+,&!\"\"F+*$)F*\"\"#\"\"\"F+F+F;F;*&F
5\"\"\"F6\"\"\"!\"\"" }}}{SECT 0 {PARA 20 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "sin(theta);" "6#-%$sinG6#%&thetaG" }{TEXT -1 1 " " }}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "isolate(vrhsub,sin(theta));" "6#
-%(isolateG6$%'vrhsubG-%$sinG6#%&thetaG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/-%$sinG6#%&thetaG*&*(&%\"vG6#%\"rG\"\"\"%\"aG\"\"\",&!\"\"F.*$)
%\"eG\"\"#F0F.F0F0*&F5\"\"\"-%%sqrtG6#*(%'lambdaGF.F/F.F1F.F0!\"\"" }}
}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "collect(subs(v[r] = dot([x/r, y
/r, z/r],[v[x], v[y], v[z]]),%),r);" "6#-%(collectG6$-%%subsG6$/&%\"vG
6#%\"rG-%$dotG6$7%*&%\"xG\"\"\"F-!\"\"*&%\"yGF4F-F5*&%\"zGF4F-F57%&F+6
#F3&F+6#F7&F+6#F9%\"%GF-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#
%&thetaG*&*(,(*&%\"xG\"\"\"&%\"vG6#F,F-F-*&%\"yGF-&F/6#F2F-F-*&%\"zGF-
&F/6#F6F-F-F-%\"aG\"\"\",&!\"\"F-*$)%\"eG\"\"#F:F-F:F:*(F?\"\"\"-%%sqr
tG6#*(%'lambdaGF-F9F-F;F-F:%\"rG\"\"\"!\"\"" }}}{EXCHG {PARA 0 "" 0 "
" {XPPEDIT 19 1 "subs(r = sqrt(x^2+y^2+z^2), %);" "6#-%%subsG6$/%\"rG-
%%sqrtG6#,(*$%\"xG\"\"#\"\"\"*$%\"yG\"\"#F/*$%\"zG\"\"#F/%\"%G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$sinG6#%&thetaG*&*(,(*&%\"xG\"\"\"&
%\"vG6#F,F-F-*&%\"yGF-&F/6#F2F-F-*&%\"zGF-&F/6#F6F-F-F-%\"aG\"\"\",&!
\"\"F-*$)%\"eG\"\"#F:F-F:F:*(F?\"\"\"-%%sqrtG6#*(%'lambdaGF-F9F-F;F-F:
-FD6#,(*$)F,F@F:F-*$)F2F@F:F-*$)F6F@F:F-F:!\"\"" }}}}{SECT 0 {PARA 20 
"" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "cos(theta);" "6#-%$cosG6#%&thet
aG" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "isolate(rh
sub,cos(theta));" "6#-%(isolateG6$%&rhsubG-%$cosG6#%&thetaG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%&thetaG*&,&*&*&%\"aG\"\"\",&!\"\"
F-*$)%\"eG\"\"#\"\"\"F-F-F4%\"rG!\"\"F-F/F-F4F2F6" }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "subs(r = sqrt(x^2+y^2+z^2),%);" "6#-%%subsG6$/%
\"rG-%%sqrtG6#,(*$%\"xG\"\"#\"\"\"*$%\"yG\"\"#F/*$%\"zG\"\"#F/%\"%G" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#%&thetaG*&,&*&*&%\"aG\"\"\"
,&!\"\"F-*$)%\"eG\"\"#\"\"\"F-F-F4*$-%%sqrtG6#,(*$)%\"xGF3F4F-*$)%\"yG
F3F4F-*$)%\"zGF3F4F-F4!\"\"F-F/F-F4F2FC" }}}}}}{SECT 0 {PARA 4 "" 0 "3
.7. argument of pericenter" {TEXT -1 27 "3.7. argument of pericenter" 
}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "rvec;" "6#%%rvecG" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/-%'matrixG6#7%7#%\"xG7#%\"yG7#%\"zG*&%\"rG\"\"
\"-F%6#7%7#,&*&-%$cosG6#%&OmegaGF0-F86#,&%&omegaGF0%&thetaGF0F0F0*(-%$
sinGF9F0-FBF<F0-F86#%\"iGF0!\"\"7#,&*&FA\"\"\"F;FKF0*(F7FKFCFKFDFKF07#
*&FCFK-FBFEF0F0" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "collect(reqs
[1]*cos(Omega)+reqs[2]*sin(Omega),r,simplify);" "6#-%(collectG6%,&*&&%
%reqsG6#\"\"\"\"\"\"-%$cosG6#%&OmegaGF,F,*&&F)6#\"\"#F,-%$sinG6#F0F,F,
%\"rG%)simplifyG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-%$cosG6#%&Om
egaG\"\"\"%\"xGF*F**&-%$sinGF(F*%\"yGF*F**&-F'6#,&%&omegaGF*%&thetaGF*
F*%\"rGF*" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "isolate(%,cos(omeg
a+theta));" "6#-%(isolateG6$%\"%G-%$cosG6#,&%&omegaG\"\"\"%&thetaGF," 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$cosG6#,&%&omegaG\"\"\"%&thetaGF)
,$*&,&*&-F%6#%&OmegaGF)%\"xGF)!\"\"*&-%$sinGF0F)%\"yGF)F3\"\"\"%\"rG!
\"\"F3" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "collect((reqs[2]*cos(
Omega)-reqs[1]*sin(Omega))*cos(i)+reqs[3]*sin(i),[r, cos(i)],simplify)
;" "6#-%(collectG6%,&*&,&*&&%%reqsG6#\"\"#\"\"\"-%$cosG6#%&OmegaGF.F.*
&&F+6#\"\"\"F.-%$sinG6#F2F.!\"\"F.-F06#%\"iGF.F.*&&F+6#\"\"$F.-F86#F=F
.F.7$%\"rG-F06#F=%)simplifyG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&-
%$cosG6#%\"iG\"\"\",&*&-F'6#%&OmegaGF*%\"yGF*F**&-%$sinGF.F*%\"xGF*!\"
\"F*F**&-F3F(F*%\"zGF*F**&-F36#,&%&omegaGF*%&thetaGF*F*%\"rGF*" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "isolate(%,sin(omega+theta));" "6
#-%(isolateG6$%\"%G-%$sinG6#,&%&omegaG\"\"\"%&thetaGF," }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%$sinG6#,&%&omegaG\"\"\"%&thetaGF),$*&,&*&-%$cos
G6#%\"iGF),&*&-F06#%&OmegaGF)%\"yGF)F)*&-F%F6F)%\"xGF)!\"\"F)F<*&-F%F1
F)%\"zGF)F<\"\"\"%\"rG!\"\"F<" }}}}{SECT 0 {PARA 4 "" 0 "3.8. Summary
" {TEXT -1 12 "3.8. Summary" }}{EXCHG {PARA 0 "" 0 "" {TEXT 262 12 "ec
centricity" }{TEXT -1 1 "\n" }{XPPEDIT 18 0 "matrix([[e[x]], [e[y]], [
e[z]]]) = matrix([[(v[y]*(x*v[y]-y*v[x])-v[z]*(z*v[x]-x*v[z]))/lambda-
x/sqrt(x^2+y^2+z^2)], [(v[z]*(y*v[z]-z*v[y])-v[x]*(x*v[y]-y*v[x]))/lam
bda-y/sqrt(x^2+y^2+z^2)], [(v[x]*(z*v[x]-x*v[z])-v[y]*(y*v[z]-z*v[y]))
/lambda-z/sqrt(x^2+y^2+z^2)]]);" "6#/-%'matrixG6#7%7#&%\"eG6#%\"xG7#&F
*6#%\"yG7#&F*6#%\"zG-F%6#7%7#,&*&,&*&&%\"vG6#F0\"\"\",&*&F,F@&F>6#F0F@
F@*&F0F@&F>6#F,F@!\"\"F@F@*&&F>6#F4F@,&*&F4F@&F>6#F,F@F@*&F,F@&F>6#F4F
@FHF@FHF@%'lambdaGFHF@*&F,F@-%%sqrtG6#,(*$F,\"\"#F@*$F0\"\"#F@*$F4\"\"
#F@FHFH7#,&*&,&*&&F>6#F4F@,&*&F0F@&F>6#F4F@F@*&F4F@&F>6#F0F@FHF@F@*&&F
>6#F,F@,&*&F,F@&F>6#F0F@F@*&F0F@&F>6#F,F@FHF@FHF@FSFHF@*&F0F@-FV6#,(*$
F,\"\"#F@*$F0\"\"#F@*$F4\"\"#F@FHFH7#,&*&,&*&&F>6#F,F@,&*&F4F@&F>6#F,F
@F@*&F,F@&F>6#F4F@FHF@F@*&&F>6#F0F@,&*&F0F@&F>6#F4F@F@*&F4F@&F>6#F0F@F
HF@FHF@FSFHF@*&F4F@-FV6#,(*$F,\"\"#F@*$F0\"\"#F@*$F4\"\"#F@FHFH" }
{TEXT -1 2 "\n\n" }{XPPEDIT 18 0 "e = sqrt(e[x]^2+e[y]^2+e[z]^2);" "6#
/%\"eG-%%sqrtG6#,(*$&F$6#%\"xG\"\"#\"\"\"*$&F$6#%\"yG\"\"#F.*$&F$6#%\"
zG\"\"#F." }{TEXT -1 2 "\n\n" }{TEXT 261 14 "semimajor axis" }{TEXT 
-1 1 "\n" }{XPPEDIT 18 0 "1/a = 2/r-v^2/lambda;" "6#/*&\"\"\"\"\"\"%\"
aG!\"\",&*&\"\"#F&%\"rGF(F&*&*$)%\"vG\"\"#F&F&%'lambdaGF(F(" }{TEXT 
-1 6 "  or  " }{XPPEDIT 18 0 "1/a = v^2/lambda-2/r;" "6#/*&\"\"\"\"\"
\"%\"aG!\"\",&*&*$)%\"vG\"\"#F&F&%'lambdaGF(F&*&\"\"#F&%\"rGF(F(" }
{TEXT -1 3 " \n\n" }{TEXT 260 11 "inclination" }{TEXT -1 1 "\n" }
{XPPEDIT 18 0 "cos(i) = (x*v[y]-y*v[x])/sqrt(lambda*a*(1-e^2));" "6#/-
%$cosG6#%\"iG*&,&*&%\"xG\"\"\"&%\"vG6#%\"yGF,F,*&F0F,&F.6#F+F,!\"\"F,-
%%sqrtG6#*(%'lambdaGF,%\"aGF,,&\"\"\"F,*$)%\"eG\"\"#F,F4F,F4" }{TEXT 
-1 6 "  or  " }{XPPEDIT 18 0 "cos(i) = (x*v[y]-y*v[x])/sqrt(lambda*a*(
e^2-1));" "6#/-%$cosG6#%\"iG*&,&*&%\"xG\"\"\"&%\"vG6#%\"yGF,F,*&F0F,&F
.6#F+F,!\"\"F,-%%sqrtG6#*(%'lambdaGF,%\"aGF,,&*$)%\"eG\"\"#\"\"\"F@\"
\"\"!\"\"\"\"\"F4" }{TEXT -1 3 " \n\n" }{TEXT 259 14 "ascending node" 
}{TEXT -1 1 "\n" }{XPPEDIT 18 0 "\{sin(Omega) = (y*v[z]-z*v[y])/(sqrt(
lambda*a*(1-e^2))*sin(i)), cos(Omega) = (x*v[z]-z*v[x])/(sqrt(lambda*a
*(1-e^2))*sin(i))\};" "6#<$/-%$sinG6#%&OmegaG*&,&*&%\"yG\"\"\"&%\"vG6#
%\"zGF-F-*&F1F-&F/6#F,F-!\"\"\"\"\"*&-%%sqrtG6#*(%'lambdaGF-%\"aGF-,&F
-F-*$)%\"eG\"\"#F6F5F-F6-F&6#%\"iG\"\"\"!\"\"/-%$cosG6#%&OmegaG*&,&*&%
\"xG\"\"\"&%\"vG6#%\"zGFQFQ*&FUFQ&FS6#FPFQ!\"\"\"\"\"*&-%%sqrtG6#*(%'l
ambdaGFZ%\"aGFZ,&\"\"\"FZ*$)%\"eG\"\"#FZ!\"\"FZFZ-%$sinG6#%\"iGFZFbo" 
}{TEXT -1 6 "  \nor\n" }{XPPEDIT 18 0 "\{sin(Omega) = (y*v[z]-z*v[y])/
(sqrt(lambda*a*(e^2-1))*sin(i)), cos(Omega) = (x*v[z]-z*v[x])/(sqrt(la
mbda*a*(e^2-1))*sin(i))\};" "6#<$/-%$sinG6#%&OmegaG*&,&*&%\"yG\"\"\"&%
\"vG6#%\"zGF-F-*&F1F-&F/6#F,F-!\"\"F-*&-%%sqrtG6#*(%'lambdaGF-%\"aGF-,
&*$%\"eG\"\"#\"\"\"\"\"\"!\"\"\"\"\"F--F&6#%\"iGF-F5/-%$cosG6#F(*&,&*&
%\"xGF-&F/6#F1F-F-*&F1F-&F/6#FOF-F5F-*&-F86#*(F;F-F<F-,&*$%\"eG\"\"#\"
\"\"\"\"\"!\"\"FDF--F&6#FGF-F5" }{TEXT -1 3 " \n\n" }{TEXT 258 12 "tru
e anomaly" }{TEXT -1 1 "\n" }{XPPEDIT 18 0 "\{sin(theta) = ((cos(Omega
)*y-sin(Omega)*x)*cos(i)+sin(i)*z)/r, cos(theta) = (cos(Omega)*x+sin(O
mega)*y)/r\};" "6#<$/-%$sinG6#%&thetaG*&,&*&,&*&-%$cosG6#%&OmegaG\"\"
\"%\"yGF2F2*&-F&6#F1F2%\"xGF2!\"\"F2-F/6#%\"iGF2F2*&-F&6#F;F2%\"zGF2F2
F2%\"rGF8/-F/6#F(*&,&*&-F/6#F1F2F7F2F2*&-F&6#F1F2F3F2F2F2F@F8" }{TEXT 
-1 25 "\nfor circular motion, or\n" }{XPPEDIT 18 0 "\{sin(theta) = (x*
v[x]+y*v[y]+z*v[z])*sqrt(a*(1-e^2))/(e*r*sqrt(lambda)), cos(theta) = a
*(1-e^2)/(e*r)-1/e\};" "6#<$/-%$sinG6#%&thetaG*(,(*&%\"xG\"\"\"&%\"vG6
#F,F-F-*&%\"yGF-&F/6#F2F-F-*&%\"zGF-&F/6#F6F-F-F--%%sqrtG6#*&%\"aGF-,&
\"\"\"F-*$%\"eG\"\"#!\"\"F-F-*(FAF-%\"rGF--F:6#%'lambdaGF-FC/-%$cosG6#
F(,&*(F=F-,&\"\"\"F-*$FA\"\"#FCF-*&FAF-FEF-FCF-*&\"\"\"F-FAFCFC" }
{TEXT -1 27 "\nfor elliptical motion, or\n" }{XPPEDIT 18 0 "\{sin(thet
a) = (x*v[x]+y*v[y]+z*v[z])*sqrt(a*(e^2-1))/(e*r*sqrt(lambda)), cos(th
eta) = a*(e^2-1)/(e*r)-1/e\};" "6#<$/-%$sinG6#%&thetaG*(,(*&%\"xG\"\"
\"&%\"vG6#F,F-F-*&%\"yGF-&F/6#F2F-F-*&%\"zGF-&F/6#F6F-F-F--%%sqrtG6#*&
%\"aGF-,&*$%\"eG\"\"#F-\"\"\"!\"\"F-F-*(F@F-%\"rGF--F:6#%'lambdaGF-FC/
-%$cosG6#F(,&*(F=F-,&*$F@\"\"#F-\"\"\"FCF-*&F@F-FEF-FCF-*&\"\"\"F-F@FC
FC" }{TEXT -1 24 "\nfor hyperbolic motion\n\n" }{TEXT 257 22 "argument
 of pericenter" }{TEXT -1 1 "\n" }{XPPEDIT 18 0 "\{sin(omega+theta) = \+
(cos(i)*(cos(Omega)*y-sin(Omega)*x)+sin(i)*z)/r, cos(omega+theta) = (c
os(Omega)*x+sin(Omega)*y)/r\};" "6#<$/-%$sinG6#,&%&omegaG\"\"\"%&theta
GF**&,&*&-%$cosG6#%\"iGF*,&*&-F06#%&OmegaGF*%\"yGF*F**&-F&6#F7F*%\"xGF
*!\"\"F*F**&-F&6#F2F*%\"zGF*F*F*%\"rGF=/-F06#,&F)F*F+F**&,&*&-F06#F7F*
F<F*F**&-F&6#F7F*F8F*F*F*FBF=" }}}}}}{MARK "3 8 0 0" 0 }{VIEWOPTS 1 1 
0 1 1 1803 }