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XXΗXXXX}XXXΗ  ($ (    ) ./0-M << deUU Y~~~~~~XSUB1~=~S~sin(~betaSUB2~-~epsilon~)~\/~sin(theta)~=~\_\_\_\_\_\_\_\_\_\_XX$}X81XXD}XX}SXX}sinXXL}(XX}82XX<}XX } XX })XX }/XX| }sinXX }(XX\ }XX })XX}XXt}_XX}_XXd}_XX}_XXT}_XX}_XXD}_XX}_XX4}_XX}_ Y~~~~~~XSUB2~=~S~sin(~betaSUB1~-~epsilon~)~\/~sin(theta)~=~\_\_\_\_\_\_\_\_\_\_ XX&X>2XXFXXSXXsinXXN(XX>1XX>XX  XX )XX /XX~ sinXX (XX^ XX )XXXXv_XX_XXf_XX_XXV_XX_XXF_XX_XX6_XX_ {~~~~~~DSUP2~=~RSUP2~+~XSUB1SUP2~-~2~R~XSUB1~cos(betaSUB1)`=~\_\_\_\_\_\_\_\_\_\_~~;~~D~=~\_\_\_\_\_\_\_\_\_\_ XX&D2XXfXX6R2XXvXXFX2U1XXXXV2XX& RXX Xn >1XX cosXX~ (XX n >1XX )XXLXX_XX_XX _XX_XX_XXt_XX_XXd_XX_XXT_XX|;XXDXXtXXD_XX_XX4_XX_XX$_XX_XX_XX_XX_XX|_ {~~~~~~DSUP2~=~RSUP2~+~XSUB2SUP2~-~2~R~XSUB2~cos(betaSUB2)`=~\_\_\_\_\_\_\_\_\_\_~~;~~D~=~\_\_\_\_\_\_\_\_\_\_ XX&D2XXfXX6R2XXvXXFX2U2XXXXV2XX& RXX Xn >2XX cosXX~ (XX n >2XX )XXLXX_XX_XX _XX_XX_XXt_XX_XXd_XX_XXT_XX|;XXDXXtXXD_XX_XX4_XX_XX$_XX_XX_XX_XX_XX|_ d<<<<'dxd Level 1 Level 2 Level 3 Level 4 Level 5("$ Figure    >/?=A<< cWPC0*}.*}93',,513.0*{ D@@@1 "}                vvv y xx w w v v  v  v   v  v  v  v   w         ! # % &~ )}" +{$ -z% .y'1x)3v, 5u/ 8t1;r4<q8=p9 ?o: @m; Al= Bk> Dj? Eh@ Eg@ FfA HdB IcC JbC KaB K_B K^B K]B J\B JZB JYB J X @ J V @ J U~ @ J T}@ JS|@ JQp@ IEo@ IEo@ IDn@ ICn@ ICn@ ICm?IBf>f>m?IBn?ICn?ICn?ICo?IDo?IDp?JEw?JNw>JOx>JPx>JQy?JRy??Sz6> Sz 5> T{7> U{7> V|7> We 6>Xd 6>Yd 5>Zd 5>[c 4>[c 4?\h@@ ]f@@ ^eAA _dAB `d AMad  BLbdBKcdBJcdBIddBHeeCHfCGg:Fh8Ei8Ej7Dk6Ck5Bl9Am7An5@o4?p3>q3=r2 =s2 <s1;t1:u1:v99w88x77y67z56{55{44|43}33~322221100 /0 ..-.----,, +, *+ )+ )* (* ') &' &' %& %& $% # % " $ " $ ! #    #    "   "                                                             DAmars_d.bmp CRight ParRight-Aligned Paragraph NumbersI.A.1.a.(1)(a)i)a)#|w (hJd$   XXXXXXXXXHP LaserJet 6P/6MP - Enhanced0 e+ :+u U(uH Z6Times New Roman RegularOC:\FPM\WPW\SNPLABS\FORWARD.GIFWPC````   !!!"""###$$$%%%&&&'''((()))***+++,,,---...///000111222333444555666777888999:::;;;<<<===>>>???@@@AAABBBCCCDDDEEEFFFGGGHHHIIIJJJKKKLLLMMMNNNOOOPPPQQQRRRSSSTTTUUUVVVWWWXXXYYYZZZ[[[\\\]]]^^^___```aaabbbcccdddeeefffggghhhiiijjjkkklllmmmnnnooopppqqqrrrssstttuuuvvvwwwxxxyyyzzz{{{|||}}}~~~ N@>C:\FPM\WPW\SNPLABS\FORWARD.GIFC:\fpm\wpw\snplabs\forward.GIF_P _z A %%)w(5hCEKQW]cioAutoList11.1.1.1.1.1.1.1.##37=CIQYag1.a.i.(1)(a)(i)1)a)(;3$2#  0  .3  0  UC:\FPM\WPW\SNPLABS\FORWARD.GIFWPC````   !!!"""###$$$%%%&&&'''((()))***+++,,,---...///000111222333444555666777888999:::;;;<<<===>>>???@@@AAABBBCCCDDDEEEFFFGGGHHHIIIJJJKKKLLLMMMNNNOOOPPPQQQRRRSSSTTTUUUVVVWWWXXXYYYZZZ[[[\\\]]]^^^___```aaabbbcccdddeeefffggghhhiiijjjkkklllmmmnnnooopppqqqrrrssstttuuuvvvwwwxxxyyyzzz{{{|||}}}~~~ T@>C:\FPM\WPW\SNPLABS\FORWARD.GIFC:\fpm\wpw\snplabs\forward.GIF_P _z A %%)w S~~~~~~theta~=~360SUP0~-~alpha~-~betaSUB1~-~betaSUB2~=~\_\_\_\_\_\_\_\_\_\_ XX&XXXX360.0XXXXXXvXXF>1XXf XX6  >2XXV XX& _XX _XX _XX _XX_XX~_XX_XXn_XX_XX^_ _~~~~~~XSUB1~=~S~CDOT~sin(~betaSUB2~-~epsilon~)~\/~sin(theta)~=~\_\_\_\_\_\_\_\_\_\_XX$}X81XXD}XX}SXX}XXn}sinXX}(XX}82XX}XX } XXf })XX6 }/XX }sinXXn }(XX }XX^})XX.}XX}_XXv}_XX}_XXf}_XX}_XXV}_XX}_XXF}_XX}_XX6}_ _~~~~~~XSUB2~=~S~CDOT~sin(~betaSUB1~-~epsilon~)~\/~sin(theta)~=~\_\_\_\_\_\_\_\_\_\_ XX&X>2XXFXXSXXXXpsinXX(XX >1XXXX  XXh )XX8 /XX sinXXp (XX XX`)XX0XX_XXx_XX_XXh_XX_XXX_XX_XXH_XX_XX8_ ~~~~~~DSUP2~=~RSUP2~+~XSUB1SUP2~-~2~CDOT~R~CDOT~XSUB1~CDOT~cos(betaSUB1)`=~\_\_\_\_\_\_\_\_\_\_~~;~~D~=~\_\_\_\_\_\_\_\_\_\_ XX&D2XXfXX6R2XXvXXFX2U1XXXXV2XX& XX RXX XX X >1XX* XX cosXX(XX >1XX\)XXXX_XX2_XX_XX"_XX_XX_XX_XX_XXz_XX_XX;XXBDXXXX_XXZ_XX_XXJ_XX_XX:_XX_XX*_XX_XX_ ~~~~~~DSUP2~=~RSUP2~+~XSUB2SUP2~-~2~CDOT~R~CDOT~XSUB2~CDOT~cos(betaSUB2)`=~\_\_\_\_\_\_\_\_\_\_~~;~~D~=~\_\_\_\_\_\_\_\_\_\_ XX&D2XXfXX6R2XXvXXFX2U2XXXXV2XX& XX RXX XX X >2XX* XX cosXX(XX >2XX\)XXXX_XX2_XX_XX"_XX_XX_XX_XX_XXz_XX_XX;XXBDXXXX_XXZ_XX_XXJ_XX_XX:_XX_XX*_XX_XX_>/?- I&mage <=8C HKKKK EJd!   XXΗXXXX}XXXΗ  _MXdXXX}  *r` *NAME_________________________________    _KEPLER_'SDETERMINATIONOFTHEDISTANCETOMARS  b     swT?/+bz0* `@E2  JJsswT?/+bz0z* `@E  JJs1.0    SetStarryNightforSeptember1,1595,12:00:01AM.Useanequatorialorientation,andturn  d offthedaylight.ThenfindMars.NoticethebrightstarAldebaraninthe VshapedfaceofTaurus,theBull,eastofMars._Centre_/LockonAldebaran.ThenrightclickonMarsandchooseTrackCelestialPath.Useatimestepof7days(solar)tostepaheadoneweekatatimeuntilyounoticeMarsundergoingretrogrademotion.EstimatethedateofthemidpointofthemotionandrecordtheJuliandatebelow.E (#(#      r Juliandate0 ______________________^  (# (# L Q5h(#RL S3   S3+25h  2  .3  0    Step35daysweeksaheadintimebeyondpart1above.RecordtheJuliandatebelow._Centre_/Lockon ` Marsandrecorditsrightascension(RA).RightclickonMarsandunselectTrackCelestialPath.FindtheSunandrecorditsRA.S3+V݌ (#(# Ќ     r Juliandate0 ______________________ (# (#    r Mars'RA0 ______________________t (# (#    r Sun'sRA0 ______________________@ (# (# 0   3.  UsetheequatorialskycharttoplotthepositionsofMarsandtheSun.Findtheshortestangular B distancealongtheeclipticbetweenMarsandtheSun.Thisangleis1.((#(#    r  ` 10 ____________o (# (# swN?/+bz0Y `@EY##JJs0   4.  Findand_Centre_/LockonMars.Setthetimestepto687days(=onesiderealperiodforMars) V"! andsteponce.RecordtheJuliandateandtheRAforMarsandtheSunasyoudidin#2above. (#(#    r Juliandate0 ______________________%>!% (# (#    r Mars'RA0 ______________________' #' (# (#    r Sun'sRA0 ______________________)$) (# (#   R+&+ _0   5.  Againuseyourequatorialskycharttofindtheshortestangulardistancealongtheecliptic b betweenMarsandtheSun.Thisangleis2.H(#(#    r  ` 20 ____________o d (# (# 0   6.  Completethecalculationstocompute,as_Kepler_Ԁdid,thedistancetoMars.  (#(# 145\LHzx *x p@Xdddddddd@E*xdL@v (#(#               (#(#1YZ\LHzx *x p@Xdddddddd@E*xddL@߰ <(#M,>e,XMXd12January2001M,~},XMXgR22October2000#MXgRX,M,~}##MXdX,M,>eE# B 1WX_OKz|x p@Xdddddddd@Exd:eL}@167_OKz|fx p@Xdddddddd@EfxdL@1[\_OKz|fx p@Xdddddddd@EfxdB>L@189\LHzx nx p@Xdddddddd@EnxdL @v1]^\LHzx nx p@Xdddddddd@EnxHddL @1:;\LHzx x p@Xdddddddd@ExdL`@v1_`\LHzx x p@Xdddddddd@ExHddL`@߰   -(. Ї@B_OKz|2 (`0*}ffff@@@E2 (jf "#(#"#(#(#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# "#(#"#(#"#(#"#(# (#(#"#(#"#(#__Ԁ=(365.25*2687)*360/365.25=42.87o  _ _Ԁ=(180__)/2=68.56o   S=2*R*sintan(__/2)=0.73185;R=1. V"!