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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 61 "Ecliptic Coordinates of t
he FAME \nViewports and Symmetry Axis" }}{PARA 19 "" 0 "" {TEXT -1 15 
"Marc A. Murison" }}{PARA 263 "" 0 "" {TEXT -1 74 "Astronomical Applic
ations Department\nU.S. Naval Observatory\nWashington, DC" }}{PARA 
266 "" 0 "" {TEXT -1 57 "murison@aa.usno.navy.mil\nhttp://aa.usno.navy
.mil/murison/" }}{PARA 265 "" 0 "" {TEXT -1 13 "19 June, 2000" }}}
{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 70 "#\n# NOTE: This document is me
ant to be read with Maple input hidden.\n#" }}}{SECT 0 {PARA 3 "" 0 "
" {TEXT -1 12 "Introduction" }}{EXCHG {PARA 264 "" 0 "" {TEXT -1 259 "
For various purposes, it is useful to have a description of the direct
ions on the sky of the FAME viewports as well as the spacecraft spin a
xis.  Given the dynamical and physical setting, the most natural coord
inate system is the ecliptic coordinate system, (" }{XPPEDIT 18 0 "bet
a;" "6#%%betaG" }{TEXT -1 1 "," }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG
" }{TEXT -1 9 "), where " }{XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT -1 
30 " is the ecliptic latitude and " }{XPPEDIT 18 0 "lambda" "6#%'lambd
aG" }{TEXT -1 153 " is the ecliptic longitude.  This Note derives the \+
ecliptic coordinates of the two viewports and the spin vector as funct
ions of a set of Euler angles, (" }{XPPEDIT 18 0 "phi;" "6#%$phiG" }
{TEXT -1 1 "," }{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 1 "," }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 195 "), where the Euler \+
angles connect the spacecraft body frame to an external (ecliptic) fra
me.  These angles and their relationship to the two coordinate frames \+
are illustrated in the figure below." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
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:::::::::::::::::::5:" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "V
iewport 1 is the leading viewport, viewport 2 is trailing.  The body f
rame coordinate system is " }{XPPEDIT 18 0 "[x, y, z];" "6#7%%\"xG%\"y
G%\"zG" }{TEXT -1 7 ".  The " }{TEXT 257 1 "y" }{TEXT -1 71 " axis coi
ncides with the viewport 1 direction.  Viewport 2 lies in the " }
{TEXT 258 2 "xy" }{TEXT -1 41 " plane and trails viewport 1 by an angl
e " }{XPPEDIT 18 0 "gamma;" "6#%&gammaG" }{TEXT -1 16 " (the so-called
 " }{TEXT 265 11 "basic angle" }{TEXT -1 8 ").  The " }{TEXT 260 1 "z
" }{TEXT -1 87 " axis is along the spacecraft symmetry axis (i.e., the
 spin axis).  The external frame " }{XPPEDIT 18 0 "[X, Y, Z];" "6#7%%
\"XG%\"YG%\"ZG" }{TEXT -1 39 " is aligned with the Sun such that the \+
" }{TEXT 261 1 "Z" }{TEXT -1 37 " axis points towards the Sun and the \+
" }{TEXT 259 2 "XZ" }{TEXT -1 47 " plane lies in the plane of the ecli
ptic.  The " }{TEXT 262 1 "Y" }{TEXT -1 47 " axis points towards the n
orth ecliptic pole.  " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 
26 " is the fast Euler angle, " }{XPPEDIT 18 0 "psi;" "6#%$psiG" }
{TEXT -1 23 " is the Sun angle, and " }{XPPEDIT 18 0 "phi;" "6#%$phiG
" }{TEXT -1 27 " is the precession angle.  " }}}}{SECT 0 {PARA 3 "" 0 
"" {TEXT -1 34 "Ecliptic Coordinates of Viewport 1" }}{EXCHG {PARA 0 "
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::::::::::::::::::3:" }{TEXT -1 0 "" }}{PARA 264 "" 0 "" {TEXT -1 22 "
In the figure above, (" }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 
1 "," }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 64 ") are eclip
tic latitude and longitude of the two viewports, and " }{XPPEDIT 18 0 
"lambda[S];" "6#&%'lambdaG6#%\"SG" }{TEXT -1 45 " is the ecliptic long
itude of the Sun.  (The " }{TEXT 264 1 "Z" }{TEXT -1 91 " axis points \+
towards the Sun.)  In this figure the spacecraft spin vector points be
low the " }{TEXT 263 2 "XZ" }{TEXT -1 89 " plane.  From the large sphe
rical triangle composed of smaller triangles A and B, we have" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "restart;" "6#%(restartG" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "eq1 := sin(beta[1]) = -sin(phi)*
sin(theta)+cos(phi)*cos(theta)*cos(psi):" "6#>%$eq1G/-%$sinG6#&%%betaG
6#\"\"\",&*&-%$sinG6#%$phiGF,-F06#%&thetaGF,!\"\"*(-%$cosG6#%$phiGF,-%
$cosG6#%&thetaGF,-F96#%$psiGF,F," }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "eq2
 := -sin(theta) = sin(beta[1])*sin(phi)+cos(beta[1])*cos(phi)*sin(lamb
da[1]-lambda[S]):" "6#>%$eq2G/,$-%$sinG6#%&thetaG!\"\",&*&-%$sinG6#&%%
betaG6#\"\"\"F4-F/6#%$phiGF4F4*(-%$cosG6#&F26#F4F4-F:6#F7F4-F/6#,&&%'l
ambdaG6#F4F4&FD6#%\"SGF+F4F4" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "eq3 := \+
cos(theta)*sin(psi) = cos(beta[1])*cos(lambda[1]-lambda[S]):" "6#>%$eq
3G/*&-%$cosG6#%&thetaG\"\"\"-%$sinG6#%$psiGF+*&-%$cosG6#&%%betaG6#F+F+
-F26#,&&%'lambdaG6#F+F+&F;6#%\"SG!\"\"F+" }}{PARA 0 "" 0 "" {XPPEDIT 
19 1 "print(eq1,`        (1)`);" "6#-%&printG6$%$eq1G%,~~~~~~~~(1)G" }
}{PARA 0 "" 0 "" {XPPEDIT 19 1 "print(eq2,`        (2)`);" "6#-%&print
G6$%$eq2G%,~~~~~~~~(2)G" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "print(eq3,` \+
       (3)`);" "6#-%&printG6$%$eq3G%,~~~~~~~~(3)G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$/-%$sinG6#&%%betaG6#\"\"\",&*&-F%6#%$phiGF*-F%6#%&theta
GF*!\"\"*(-%$cosGF.F*-F6F1F*-F66#%$psiGF*F*%,~~~~~~~~(1)G" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6$/,$-%$sinG6#%&thetaG!\"\",&*&-F&6#&%%betaG6#\"
\"\"F1-F&6#%$phiGF1F1*(-%$cosGF-F1-F7F3F1-F&6#,&&%'lambdaGF0F1&F=6#%\"
SGF)F1F1%,~~~~~~~~(2)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/*&-%$cosG6#
%&thetaG\"\"\"-%$sinG6#%$psiGF)*&-F&6#&%%betaG6#F)F)-F&6#,&&%'lambdaGF
3F)&F86#%\"SG!\"\"F)%,~~~~~~~~(3)G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 37 "Putting eq. (1) into eq. (2), we find" }}}{EXCHG {PARA 0 "" 0 "
" {XPPEDIT 19 1 "sinfix(eqcollect(subs(eq1,eq2))):" "6#-%'sinfixG6#-%*
eqcollectG6#-%%subsG6$%$eq1G%$eq2G" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "e
q4 := eqcollect(%):" "6#>%$eq4G-%*eqcollectG6#%\"%G" }}{PARA 0 "" 0 "
" {XPPEDIT 19 1 "print(eq4,`        (4)`);" "6#-%&printG6$%$eq4G%,~~~~
~~~~(4)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/,$*&-%$sinG6#%&thetaG\"\"
\"-%$cosG6#%$phiGF*!\"\",&*(-F,6#%$psiGF*-F,F(F*-F'F-F*F**&-F'6#,&&%'l
ambdaG6#F*F*&F<6#%\"SGF/F*-F,6#&%%betaGF=F*F*%,~~~~~~~~(4)G" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Now,  " }{XPPEDIT 18 0 "0 < cos(bet
a[1]);" "6#2\"\"!-%$cosG6#&%%betaG6#\"\"\"" }{TEXT -1 9 " for all " }
{XPPEDIT 18 0 "beta[1];" "6#&%%betaG6#\"\"\"" }{TEXT -1 53 ".  We may \+
therefore make explicit use of (1) and set " }{XPPEDIT 18 0 "cos(beta[
1]) = sqrt(1-sin(beta[1])^2);" "6#/-%$cosG6#&%%betaG6#\"\"\"-%%sqrtG6#
,&F*F**$-%$sinG6#&F(6#F*\"\"#!\"\"" }{TEXT -1 43 " in (3) and (4), obt
aining expressions for " }{XPPEDIT 18 0 "lambda[1];" "6#&%'lambdaG6#\"
\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "eq5 := \+
subs(cos(beta[1]) = sqrt(1-rhs(eq1)^2),isolate(eq3,op(select(has,indet
s(eq3,function),lambda[1])))):" "6#>%$eq5G-%%subsG6$/-%$cosG6#&%%betaG
6#\"\"\"-%%sqrtG6#,&F/F/*$-%$rhsG6#%$eq1G\"\"#!\"\"-%(isolateG6$%$eq3G
-%#opG6#-%'selectG6%%$hasG-%'indetsG6$F>%)functionG&%'lambdaG6#F/" }}
{PARA 0 "" 0 "" {XPPEDIT 19 1 "eq6 := subs(cos(beta[1]) = sqrt(1-rhs(e
q1)^2),isolate(eq4,op(select(has,indets(eq4,function),lambda[1])))):" 
"6#>%$eq6G-%%subsG6$/-%$cosG6#&%%betaG6#\"\"\"-%%sqrtG6#,&F/F/*$-%$rhs
G6#%$eq1G\"\"#!\"\"-%(isolateG6$%$eq4G-%#opG6#-%'selectG6%%$hasG-%'ind
etsG6$F>%)functionG&%'lambdaG6#F/" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "pr
int(eq5,`        (5)`);" "6#-%&printG6$%$eq5G%,~~~~~~~~(5)G" }}{PARA 
0 "" 0 "" {XPPEDIT 19 1 "print(eq6,`        (6)`);" "6#-%&printG6$%$eq
6G%,~~~~~~~~(6)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%$cosG6#,&&%'lam
bdaG6#\"\"\"F+&F)6#%\"SG!\"\"*&*&-F%6#%&thetaGF+-%$sinG6#%$psiGF+F+*$-
%%sqrtG6#,&F+F+*$),&*&-F66#%$phiGF+-F6F3F+F/*(-F%FCF+F2F+-F%F7F+F+\"\"
#F+F/F+F/%,~~~~~~~~(5)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%$sinG6#,
&&%'lambdaG6#\"\"\"F+&F)6#%\"SG!\"\",$*&,&*&-F%6#%&thetaGF+-%$cosG6#%$
phiGF+F+*(-F86#%$psiGF+-F8F5F+-F%F9F+F+F+*$-%%sqrtG6#,&F+F+*$),&*&F@F+
F4F+F/*(F7F+F?F+F<F+F+\"\"#F+F/F+F/F/%,~~~~~~~~(6)G" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 10 "Expanding " }{XPPEDIT 18 0 "sin(lambda[1]-lambd
a[S]);" "6#-%$sinG6#,&&%'lambdaG6#\"\"\"F*&F(6#%\"SG!\"\"" }{TEXT -1 
5 " and " }{XPPEDIT 18 0 "cos(lambda[1]-lambda[S]);" "6#-%$cosG6#,&&%'
lambdaG6#\"\"\"F*&F(6#%\"SG!\"\"" }{TEXT -1 190 " in (5) and (6), and \+
carrying along (1), we arrive at a prescription for determining the ec
liptic coordinates of viewport 1 as a function of the Euler angles and
 of the longitude of the Sun:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 
1 "\{seq(subs(lhs(eq || (k+4)) = expand(lhs(eq || (k+4))),rhs(eq || (k
+4)) = A || k,eq || (k+4)),k = 1 .. 2)\}:" "6#<#-%$seqG6$-%%subsG6%/-%
$lhsG6#(%#eqG,&%\"kG\"\"\"\"\"%F2-%'expandG6#-F,6#(F/,&F1F2F3F2/-%$rhs
G6#(F/,&F1F2F3F2(%\"AGF1(F/,&F1F2F3F2/F1;F2\"\"#" }}{PARA 0 "" 0 "" 
{XPPEDIT 19 1 "solve(%,\{cos(lambda[1]), sin(lambda[1])\}):" "6#-%&sol
veG6$%\"%G<$-%$cosG6#&%'lambdaG6#\"\"\"-%$sinG6#&F,6#F." }}{PARA 0 "" 
0 "" {XPPEDIT 19 1 "subs(seq(A || k = rhs(eq || (k+4)),k = 1 .. 2),%):
" "6#-%%subsG6$-%$seqG6$/(%\"AG%\"kG-%$rhsG6#(%#eqG,&F,\"\"\"\"\"%F3/F
,;F3\"\"#%\"%G" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "convert(select(hastyp
e,indets(%),sqrt),list):" "6#-%(convertG6$-%'selectG6%%(hastypeG-%'ind
etsG6#%\"%G%%sqrtG%%listG" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "zip(proc (
x, y) options operator, arrow; x = y end proc,%,[seq(A || k,k = 1 .. n
ops(%))]):" "6#-%$zipG6%R6$%\"xG%\"yG7\"6$%)operatorG%&arrowG6\"/F(F)F
.F.F.%\"%G7#-%$seqG6$(%\"AG%\"kG/F7;\"\"\"-%%nopsG6#F0" }}{PARA 0 "" 
0 "" {XPPEDIT 19 1 "subs(%,%%%):" "6#-%%subsG6$%\"%G%$%%%G" }}{PARA 0 
"" 0 "" {XPPEDIT 19 1 "collect(%,[seq(A || k,k = 1 .. nops(%))]):" "6#
-%(collectG6$%\"%G7#-%$seqG6$(%\"AG%\"kG/F-;\"\"\"-%%nopsG6#F&" }}
{PARA 0 "" 0 "" {XPPEDIT 19 1 "invsubs(%%%,%):" "6#-%(invsubsG6$%$%%%G
%\"%G" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "eq7 := op(select(has,%,cos(lam
bda[1]))):" "6#>%$eq7G-%#opG6#-%'selectG6%%$hasG%\"%G-%$cosG6#&%'lambd
aG6#\"\"\"" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "eq8 := op(select(has,%%,s
in(lambda[1]))):" "6#>%$eq8G-%#opG6#-%'selectG6%%$hasG%#%%G-%$sinG6#&%
'lambdaG6#\"\"\"" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "print(eq1,`        \+
(1)`);" "6#-%&printG6$%$eq1G%,~~~~~~~~(1)G" }}{PARA 0 "" 0 "" 
{XPPEDIT 19 1 "print(eq7,`        (7)`);" "6#-%&printG6$%$eq7G%,~~~~~~
~~(7)G" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "print(eq8,`        (8)`);" "6
#-%&printG6$%$eq8G%,~~~~~~~~(8)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-
%$sinG6#&%%betaG6#\"\"\",&*&-F%6#%$phiGF*-F%6#%&thetaGF*!\"\"*(-%$cosG
F.F*-F6F1F*-F66#%$psiGF*F*%,~~~~~~~~(1)G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$/-%$cosG6#&%'lambdaG6#\"\"\"*&,&*(-F%6#&F(6#%\"SGF*-F%6#%&thetaG
F*-%$sinG6#%$psiGF*F**&,&*&-F7F4F*-F%6#%$phiGF*F**(-F%F8F*F3F*-F7F?F*F
*F*-F7F/F*F*F**$-%%sqrtG6#,&F*F**$),&*&FCF*F=F*!\"\"*(F>F*F3F*FBF*F*\"
\"#F*FNF*FN%,~~~~~~~~(7)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%$sinG6
#&%'lambdaG6#\"\"\"*&,&*(-%$cosG6#%&thetaGF*-F%6#%$psiGF*-F%6#&F(6#%\"
SGF*F**&-F/F6F*,&*&-F%F0F*-F/6#%$phiGF*F**(-F/F3F*F.F*-F%F@F*F*F*!\"\"
F**$-%%sqrtG6#,&F*F**$),&*&FDF*F>F*FE*(F?F*F.F*FCF*F*\"\"#F*FEF*FE%,~~
~~~~~~(8)G" }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 76 "One can view this
 as a coordinate transformation, given values of Sun angle " }
{XPPEDIT 18 0 "psi;" "6#%$psiG" }{TEXT -1 21 " and solar longitude " }
{XPPEDIT 18 0 "lambda[S];" "6#&%'lambdaG6#%\"SG" }{TEXT -1 31 " , from
 the Euler coordinates (" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT 
-1 1 "," }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 31 ") to the eclip
tic coordinates (" }{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 1 "," 
}{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 121 ") of viewport 1.
  The transformation is undefined at the points where the denominators
 of (7) and (8) are zero.  Setting " }{XPPEDIT 18 0 "sqrt(1-(-sin(phi)
*sin(theta)+cos(phi)*cos(theta)*cos(psi))^2) = 0;" "6#/-%%sqrtG6#,&\"
\"\"F(*$),&*&-%$sinG6#%$phiGF(-F.6#%&thetaGF(!\"\"*(-%$cosGF/F(-F7F2F(
-F76#%$psiGF(F(\"\"#F(F4\"\"!" }{TEXT -1 40 ", we find the following e
ight solutions:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "\{solve(1-(-
sin(phi)*sin(theta)+cos(phi)*cos(theta)*cos(psi))^2)\}:" "6#<#-%&solve
G6#,&\"\"\"F(*$,&*&-%$sinG6#%$phiGF(-F-6#%&thetaGF(!\"\"*(-%$cosG6#F/F
(-F66#F2F(-F66#%$psiGF(F(\"\"#F3" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "rem
ove(has,`minus`(indets(%,name),\{Pi, psi, phi, theta\}),%):" "6#-%'rem
oveG6%%$hasG-%&minusG6$-%'indetsG6$%\"%G%%nameG<&%#PiG%$psiG%$phiG%&th
etaGF-" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "convert(map(proc (x) options \+
operator, arrow; x = 0 end proc,%),list):" "6#-%(convertG6$-%$mapG6$R6
#%\"xG7\"6$%)operatorG%&arrowG6\"/F+\"\"!F0F0F0%\"%G%%listG" }}{PARA 
0 "" 0 "" {XPPEDIT 19 1 "subs(%,%%%):" "6#-%%subsG6$%\"%G%$%%%G" }}
{PARA 0 "" 0 "" {XPPEDIT 19 1 "sinfix(cosfix(rootfunc(%,simplify))):" 
"6#-%'sinfixG6#-%'cosfixG6#-%)rootfuncG6$%\"%G%)simplifyG" }}{PARA 0 "
" 0 "" {XPPEDIT 19 1 "subs(sqrt(sin(phi)^2*cos(theta)^2) = abs(sin(phi
))*abs(cos(theta)),%):" "6#-%%subsG6$/-%%sqrtG6#*&-%$sinG6#%$phiG\"\"#
-%$cosG6#%&thetaGF/*&-%$absG6#-F,6#F.\"\"\"-F66#-F16#F3F:%\"%G" }}
{PARA 0 "" 0 "" {XPPEDIT 19 1 "convert(map(proc (x) options operator, \+
arrow; `minus`(x,\{phi = phi, psi = psi, theta = theta\}) end proc,%),
list):" "6#-%(convertG6$-%$mapG6$R6#%\"xG7\"6$%)operatorG%&arrowG6\"-%
&minusG6$F+<%/%$phiGF6/%$psiGF8/%&thetaGF:F0F0F0%\"%G%%listG" }}{PARA 
0 "" 0 "" {XPPEDIT 19 1 "denomsols := sort_by_cost(%):" "6#>%*denomsol
sG-%-sort_by_costG6#%\"%G" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "for p in d
enomsols do print(op(p)) end do;" "6#?&%\"pG%*denomsolsG%%trueG-%&prin
tG6#-%#opG6#F$" }}{PARA 0 "" 0 "" {XPPEDIT 19 1 "p := 'p':" "6#>%\"pG.
F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%&thetaG\"\"!/%$psiG-%'arccosG6
#*&\"\"\"F,-%$cosG6#%$phiG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%&
thetaG%#PiG/%$psiG-%'arccosG6#*&\"\"\"F,-%$cosG6#%$phiG!\"\"" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$/%&thetaG%#PiG/%$psiG,&F%\"\"\"-%'arccosG6#*
&F)F)-%$cosG6#%$phiG!\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%&thet
aG\"\"!/%$psiG,&%#PiG\"\"\"-%'arccosG6#*&F*F*-%$cosG6#%$phiG!\"\"F3" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$psiG-%'arccosG6#*&,&\"\"\"F**&-%$s
inG6#%$phiGF*-F-6#%&thetaGF*!\"\"F**&-%$cosGF1F*-F6F.F*F3" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/%$psiG-%'arccosG6#*&,&!\"\"\"\"\"*&-%$sinG6#%$
phiGF+-F.6#%&thetaGF+F+F+*&-%$cosGF2F+-F6F/F+F*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%$psiG-%'arccosG6#*&,&!\"\"\"\"\"*&-%$sinG6#%$phiGF+-F
.6#%&thetaGF+F*F+*&-%$cosGF2F+-F6F/F+F*" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/%$psiG-%'arccosG6#*&,&\"\"\"F**&-%$sinG6#%$phiGF*-F-6#%&thetaGF
*F*F**&-%$cosGF1F*-F5F.F*!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
102 "These solutions are satisfied for real values of the parameters a
t only the following specific points:" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 19 1 "solve(arccos(1/cos(phi)) = Pi);" "6#-%&solveG6#/-%'arcc
osG6#*&\"\"\"F+-%$cosG6#%$phiG!\"\"%#PiG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,&%#PiG\"\"\"*(\"\"#F%F$F%%&_Z23|irGF%F%" }}}{EXCHG {PARA 0 "" 
0 "" {XPPEDIT 19 1 "zeros := [[theta = 0, psi = 0, phi = 0], [theta = \+
0, psi = Pi, phi = Pi], [theta = Pi, psi = 0, phi = 0], [theta = Pi, p
si = Pi, phi = Pi], [theta = 0, psi = Pi, phi = 0], [theta = 0, psi = \+
0, phi = Pi], [theta = Pi, psi = Pi, phi = 0], [theta = Pi, psi = 0, p
hi = Pi]]:" "6#>%&zerosG7*7%/%&thetaG\"\"!/%$psiGF)/%$phiGF)7%/F(F)/F+
%#PiG/F-F17%/F(%#PiG/F+F)/F-F)7%/F(%#PiG/F+F1/F-F17%/F(F)/F+F1/F-F)7%/
F(F)/F+F)/F-F17%/F(%#PiG/F+F1/F-F)7%/F(%#PiG/F+F)/F-F1" }}}{EXCHG 
{PARA 0 "" 0 "" {XPPEDIT 19 1 "op(zeros);" "6#-%#opG6#%&zerosG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6*7%/%&thetaG\"\"!/%$psiGF&/%$phiGF&7%F$
/F(%#PiG/F*F-7%/F%F-F'F)7%F0F,F.7%F$F,F)7%F$F'F.7%F0F,F)7%F0F'F." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "The corresponding values of the ec
liptic latitude " }{XPPEDIT 18 0 "beta[1];" "6#&%%betaG6#\"\"\"" }
{TEXT -1 4 " are" }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 1 0 280 "Fextract
 := proc( eqn::equation )\n  local vars, expr;\n  expr := rhs(eqn);\n \+
 vars := indets(expr,name);\n  isolate(eqn,op(indets(lhs(eqn),name)));
\n  remove(has,indets(%,name) minus indets(lhs(eqn),name) minus \{Pi\}
,vars);\n  convert(map(x->x=0,%),list);\n  simp(subs(%,%%%));\nend pro
c:" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "\{seq(Fextract(eval(subs(
zeros[k],eq1))),k = 1 .. nops(zeros))\};" "6#<#-%$seqG6$-%)FextractG6#
-%%evalG6#-%%subsG6$&%&zerosG6#%\"kG%$eq1G/F3;\"\"\"-%%nopsG6#F1" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#<$/&%%betaG6#\"\"\",$%#PiG#F(\"\"#/F%,
$F*#!\"\"F," }}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 127 "So the trouble \+
spots are at the ecliptic poles, as one might expect.  In the plot bel
ow, the red curves are curves of constant " }{XPPEDIT 18 0 "phi;" "6#%
$phiG" }{TEXT -1 21 " (evenly spaced), as " }{XPPEDIT 18 0 "theta;" "6
#%&thetaG" }{TEXT -1 21 " varies in the range " }{XPPEDIT 18 0 "[0, 2*
Pi];" "6#7$\"\"!*&\"\"#\"\"\"%#PiGF'" }{TEXT -1 51 ".  The circular bl
ue curves are curves of constant " }{XPPEDIT 18 0 "theta;" "6#%&thetaG
" }{TEXT -1 39 " (again, for evenly-spaced values), as " }{XPPEDIT 18 
0 "phi;" "6#%$phiG" }{TEXT -1 21 " varies in the range " }{XPPEDIT 18 
0 "[0, Pi/2];" "6#7$\"\"!*&%#PiG\"\"\"\"\"#!\"\"" }{TEXT -1 16 ".  The
 abscissa " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{XPPEDIT 18 0 "lambd
a = lambda-lambda[S];" "6#/%'lambdaG,&F$\"\"\"&F$6#%\"SG!\"\"" }{TEXT 
-1 32 " , and the Sun angle was set to " }{XPPEDIT 18 0 "psi = Pi/4;" 
"6#/%$psiG*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 208 ".  The zones of avoi
dance show in a graphic way the need for observations throughout the y
ear, so that the holes slide along in longitude and, over a long enoug
h time, the sky coverage is somewhat evened out." }}}{EXCHG {PARA 0 "
" 0 "" {XPPEDIT 19 1 "with(plotting):" "6#-%%withG6#%)plottingG" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Fbeta1 := fn(rhs(Fextract(eq1)),
theta,phi,psi):" "6#>%'Fbeta1G-%#fnG6&-%$rhsG6#-%)FextractG6#%$eq1G%&t
hetaG%$phiG%$psiG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Fclambda1 \+
:= fn(cos(rhs(Fextract(eq7))),theta,phi,psi,lambda[S]):" "6#>%*Fclambd
a1G-%#fnG6'-%$cosG6#-%$rhsG6#-%)FextractG6#%$eq7G%&thetaG%$phiG%$psiG&
%'lambdaG6#%\"SG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "Fslambda1 :
= fn(sin(rhs(Fextract(eq8))),theta,phi,psi,lambda[S]):" "6#>%*Fslambda
1G-%#fnG6'-%$sinG6#-%$rhsG6#-%)FextractG6#%$eq8G%&thetaG%$phiG%$psiG&%
'lambdaG6#%\"SG" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 19 1 "coordplot2D(
[arctan(Fslambda1(theta,phi,Pi/4,0),Fclambda1(theta,phi,Pi/4,0))/deg, \+
Fbeta1(theta,phi,Pi/4)/deg],theta = 0 .. 2*Pi,phi = 0 .. 2*Pi,21,style
 = point,symbol = circle,symbolsize = 4,numpoints = 500,title = \"ecli
ptic coordinates of viewport 1\",labels = [\"Dl\", \"b\"],labelfont = \+
[SYMBOL, 20],view = [-180 .. 180, -90 .. 90]);" "6#-%,coordplot2DG6.7$
*&-%'arctanG6$-%*Fslambda1G6&%&thetaG%$phiG*&%#PiG\"\"\"\"\"%!\"\"\"\"
!-%*Fclambda1G6&F.F/*&F1F2F3F4F5F2%$degGF4*&-%'Fbeta1G6%F.F/*&F1F2F3F4
F2F:F4/F.;F5*&\"\"#F2F1F2/F/;F5*&FCF2F1F2\"#@/%&styleG%&pointG/%'symbo
lG%'circleG/%+symbolsizeGF3/%*numpointsG\"$+&/%&titleGQCecliptic~coord
inates~of~viewport~16\"/%'labelsG7$Q#DlFVQ\"bFV/%*labelfontG7$%'SYMBOL
G\"#?/%%viewG7$;,$\"$!=F4F`o;,$\"#!*F4Fco" }}{PARA 13 "" 1 "" 
{GLPLOT2D 691 471 471 {PLOTDATA 2 "6V-%'CURVESG6'7`jl7$$\"\"!F)$\"3s++
+++++X!#;7$$!3E%>cK$z`Zv!#=$\"3#[N4i#G]*\\%F,7$$!3V8/%=Jb6T\"!#<$\"3s
\"eb!3:E)\\%F,7$$!3+z)p03r'[@F6$\"3=sc@6t'f\\%F,7$$!3)R*\\>uh***)GF6$
\"3'Hl1!\\!*p#\\%F,7$$!3Sj'Rjecji$F6$\"3g!)p\"*3L\\)[%F,7$$!346\"y%))p
X2VF6$\"3M-J?Eru$[%F,7$$!30J>I(\\j2,&F6$\"3#=@<@$p(zZ%F,7$$!30*3k9IFdt
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