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GHA B" VY cos D = sin Dec M sin Dec B cos Dec M cos Dec Bcos [15 RA M [h ! RA B [h " 8 QZ [PM OMVKMU[YQK S\UIY LQZ[IUKM& CPMZM NVYT\SIZ KIU JM \ZML [V ZM[ \W VUMcZ V^U [IJSM ^Q[P [PM IQL VN [PM ?I\[QKIS 5STIUIK VY IU` KVTW\[MY ISTIUIK QN I S\UIY LQZ[IUKM [IJSM QZ UV[ I]IQSIJSM& "5STVZ[ I KMU[\Y` IN[MY [PM VYQOQUIS =\UIY 8QZ[IUKM CIJSMZ ^MYM LYVWWML$ B[M]MU EMWZ[MY YMZ\TML [PM [YILQ[QVU& ;QZ [IJSMZ IYM WYMZMU[S` *((,! I]IQSIJSM [PYV\OP [PM QU[MYUM[ G),H& *951>8;6 @75 9A;1> 48?@1;35 6MNVYM I S\UIY LQZ[IUKM TMIZ\YML J` [PM VJZMY]MY KIU JM KVTWIYML ^Q[P [IJ\SI[ML ]IS\MZ$ Q[ PIZ [V JM YML\KML [V [PM KVYYMZWVULQUO OMVKMU[YQK IUOSM J` KSMIYQUO Q[ NYVT [PM MNNMK[Z VN YMNYIK[QVU IUL WIYISSI_& CPQZ MZZMU[QIS WYVKMZZ QZ KISSML 3951>8;6 @75 9A;1> 48?@1;35 & ?\TMYV\Z WYVKML\YMZ PI]M JMMU LM]MSVWML$ ITVUO [PMT YQOVYV\Z IUL dX\QKRe TM[PVLZ& /%) :QO& /%) ZPV^Z [PM WVZQ[QVUZ VN [PM TVVU$ >$ IUL I YMNMYMUKM JVL`$ 6$ QU [PM KVVYLQUI[M Z`Z[MT VN [PM PVYQaVU& EM LMUV[M [PM IWWIYMU[ WVZQ[QVUZ VN [PM KMU[MYZ VN [PM TVVU IUL [PM YMNMYMUKM JVL` J` >IWW IUL 6IWW $ YMZWMK[Q]MS`& F QZ [PM aMUQ[P& CPM ZQLM 8IWW VN [PM ZWPMYQKIS [YQIUOSM 6IWW %F%> IWW QZ [PM IWWIYMU[ S\UIY LQZ[IUKM& CPM IS[Q[\LMZ VN >IWW IUL 6IWW VJ[IQUML IN[MY IWWS`QUO [PM KVYYMK[QVUZ NVY QULM_ MYYVY$ LQW$ IUL ZMTQLQITM[MY! 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Az B CPM WVZQ[QVU VN MIKP JVL` QZ ZPQN[ML ISVUO Q[Z ]MY[QKIS KQYKSM J` I[TVZWPMYQK YMNYIK[QVU IUL WIYISSI_ QU IS[Q[\LM& 5N[MY KVYYMK[QUO ;>IWW IUL;6IWW NVY JV[P MNNMK[Z$ ^M VJ[IQU [PM OMVKMU[YQK WVZQ[QVUZ > IUL 6& EM LMUV[M [PM IS[Q[\LM VN > J` ;> IUL [PM IS[Q[\LM VN 6 J` ;6&;> QZ IS^I`Z OYMI[MY [PIU ;>IWW JMKI\ZM [PM WIYISSI_ VN [PM TVVU QZ IS^I`Z OYMI[MY [PIU YMNYIK[QVU& CPM IUOSM QZ UMQ[PMY INNMK[ML J` YMNYIK[QVU UVY J` [PM WIYISSI_ QU IS[Q[\LM2 Az M = Az Mapp Az B = Az Bapp CPM ZQLM 8 VN [PM ZWPMYQKIS [YQIUOSM 6%F%> QZ [PM \URUV^U OMVKMU[YQK S\UIY LQZ[IUKM& 5WWS`QUO [PM KVZQUM NVYT\SI [V [PM ZWPMYQKIS [YQIUOSM NVYTML J` [PM aMUQ[P IUL [PM IWWIYMU[ WVZQ[QVUZ$ ^M OM[2 cos Dapp = sin H Mapp sin H Bapp cos H Mapp cos H Bapp cos # cos D ! sin H sin H cos # = app Mapp Bapp cos H Mapp cos H Bapp AMWMI[QUO [PM WYVKML\YM ^Q[P [PM ZWPMYQKIS [YQIUOSM NVYTML J` [PM aMUQ[P IUL [PM OMVKMU[YQK WVZQ[QVUZ$ ^M OM[2 cos D = sin H M sin H B cos H M cos H Bcos # cos D ! sin H sin H cos # = M B cos H M cos H B /%* BQUKM QZ KVUZ[IU[$ ^M KIU KVTJQUM JV[P IaQT\[P NVYT\SIZ2 cos D ! sin H sin H cos D ! sin H sin H M B = app Mapp Bapp cos H M cos H B cos H Mapp cos H Bapp CP\Z$ ^M PI]M MSQTQUI[ML [PM \URUV^U IUOSM & ?V^$ ^M Z\J[YIK[ \UQ[` NYVT JV[P ZQLMZ VN [PM MX\I[QVU2 cos D ! sin H sin H cos D ! sin H sin H M B ! 1 = app Mapp Bapp ! 1 cos H M cos H B cos H Mapp cos H Bapp cos D ! sin H sin H cos H cos H cos D ! sin H sin H cos H cos H M B ! M B = app Mapp Bapp ! Mapp Bapp cos H M cos H B cos H M cos H B cos H Mapp cos H Bapp cos H Mapp cos H Bapp cos D ! sin H sin H ! cos H cos H cos D ! sin H sin H ! cos H cos H M B M B = app Mapp Bapp Mapp Bapp cos H M cos H B cos H Mapp cos H Bapp DZQUO [PM ILLQ[QVU NVYT\SI NVY KVZQUMZ$ ^M PI]M2 cos D ! cos H ! H " cos D ! cos H ! H " M B = app Mapp Bapp cos H M cos H B cos H Mapp cos H Bapp BVS]QUO NVY KVZ 8$ ^M VJ[IQU D2-1)./-'30 NVYT\SI NVY KSMIYQUO [PM S\UIY LQZ[IUKM2 cos H M cos H B cos D = [cos D app ! cos H Mapp ! H Bapp " cos H M ! H B" cos H Mapp cos H Bapp 5LLQUO \UQ[` [V JV[P ZQLMZ VN [PM MX\I[QVU QUZ[MIL VN Z\J[YIK[QUO Q[$ SMILZ [V $.2-(30 NVYT\SI2 cos H M cos H B cos D = [ cos Dapp cos H Mapp H Bapp " ! cos H M H B" cos H Mapp cos H Bapp /><354A>5 8MYQ]QUO DC NYVT I S\UIY LQZ[IUKM KVTWYQZMZ [PM NVSSV^QUO Z[MWZ2 . EM TMIZ\YM [PM IS[Q[\LM VN [PM \WWMY VY SV^MY SQTJ VN [PM TVVU$ ^PQKPM]MY QZ ]QZQJSM$ IUL UV[M [PM [QTM VN [PM VJZMY]I[QVU QULQKI[ML J` V\Y ^I[KP$ EC) =>IWW & EM IWWS` [PM KVYYMK[QVUZ NVY QULM_ MYYVY IUL LQW QN UMKMZZIY`! IUL OM[ [PM IWWIYMU[ IS[Q[\LM VN [PM SQTJ$ ;) =>IWW & EM YMWMI[ [PM WYVKML\YM ^Q[P [PM YMNMYMUKM JVL` IUL VJ[IQU [PM ^I[KP [QTM EC) 6IWW IUL [PM IS[Q[\LM ;) 6IWW & !. EM TMIZ\YM [PM IUO\SIY LQZ[IUKM JM[^MMU [PM SQTJ VN [PM TVVU IUL [PM YMNMYMUKM JVL`$ 8=IWW $ IUL UV[M [PM KVYYMZWVULQUO ^I[KP [QTM$ EC 8& CPM IUOSM 8=IWW PIZ [V JM TMIZ\YML ^Q[P [PM OYMI[MZ[ WVZZQJSM WYMKQZQVU& <[ QZ YMKVTTMULML [V TMIZ\YM I NM^ 8=IWW ]IS\MZ IUL [PMQY KVYYMZWVULQUO EC 8 ]IS\MZ QU YIWQL Z\KKMZZQVU IUL KISK\SI[M [PM YMZWMK[Q]M I]MYIOM ]IS\M& EPMU [PM TVVU QZ ISTVZ[ N\SS$ Q[ QZ UV[ X\Q[M MIZ` [V LQZ[QUO\QZP [PM SQTJ VN [PM TVVU NYVT [PM [MYTQUI[VY ZPILV^ SQUM!& /%+ ". EM TMIZ\YM [PM IS[Q[\LMZ VN JV[P JVLQMZ IOIQU$ IZ LMZKYQJML IJV]M& EM LMUV[M [PMT J` ;* =>IWW IUL ;* 6IWW $ IUL UV[M [PM KVYYMZWVULQUO ^I[KP [QTMZ VN VJZMY]I[QVU$ EC* =>IWW IUL EC* 6IWW & #. BQUKM [PM VJZMY]I[QVUZ IYM VUS` I NM^ TQU\[MZ IWIY[$ ^M KIU KISK\SI[M [PM IS[Q[\LM VN [PM YMZWMK[Q]M JVL` I[ [PM TVTMU[ VN [PM S\UIY LQZ[IUKM VJZMY]I[QVU J` SQUMIY QU[MYWVSI[QVU2 WT D ! WT1 LMapp H LMapp = H1 LMapp H2 LMapp ! H1 LMapp " WT2 LMapp ! WT1 LMapp WT D ! WT1 Bapp H Bapp = H1 Bapp H2 Bapp ! H1 Bapp " WT2 Bapp ! WT1 Bapp $. EM KVYYMK[ [PM IS[Q[\LM VN [PM TVVU IUL [PM IUO\SIY LQZ[IUKM 8=IWW NVY [PM I\OTMU[ML ZMTQLQITM[MY VN [PM TVVU$ B8 I\O & CPM SI[[MY KIU JM KISK\SI[ML LQYMK[S` NYVT [PM IS[Q[\LM VN [PM \WWMY VY SV^MY SQTJ VN [PM TVVU2 k tan SD aug = k = 0.2725 1 ! cos H ! k"2 ! sin H sin 2 HP LMapp LMapp [$ M upper limb: cos H LMapp ! k lower limb: cos H LMapp k CPM IS[Q[\LM KVYYMK[QVU QZ2 Lower limb: H Mapp = H LMapp SD aug Upper limb: H Mapp = H LMapp ! SD aug CPM Y\SMZ NVY [PM S\UIY LQZ[IUKM KVYYMK[QVU IYM2 Limb of moon towards reference body: Dapp = DLapp SD aug Limb of moon away from reference body: Dapp = DLapp ! SD aug CPM IJV]M WYVKML\YM QZ IU IWWYV_QTI[QVU ZQUKM [PM I\OTMU[ML ZMTQLQITM[MY QZ I N\UK[QVU VN [PM IS[Q[\LM KVYYMK[ML NVY YMNYIK[QVU& BQUKM YMNYIK[QVU QZ I ZTISS X\IU[Q[` IUL ZQUKM [PM [V[IS I\OTMU[I[QVU JM[^MMU (f IUL 1(f IS[Q[\LM QZ VUS` IWWYV_& (&+c$ [PM YMZ\S[QUO MYYVY QZ ]MY` ZTISS IUL TI` JM QOUVYML& CPM Z\U$ ^PMU KPVZMU IZ YMNMYMUKM JVL`$ YMX\QYMZ [PM ZITM KVYYMK[QVUZ NVY ZMTQLQITM[MY& BQUKM [PM Z\U LVMZ UV[ ZPV^ I TMIZ\YIJSM I\OTMU[I[QVU$ ^M KIU \ZM [PM OMVKMU[YQK ZMTQLQITM[MY [IJ\SI[ML QU [PM ?I\[QKIS 5STIUIK VY KISK\SI[ML ^Q[P I KVTW\[MY WYVOYIT& %. EM KVYYMK[ JV[P IS[Q[\LMZ$ ; >IWW IUL ; 6IWW $ NVY I[TVZWPMYQK YMNYIK[QVU$ A& p[mbar 283 0.97127 0.00137 R [' = ! i = Mapp , Bapp H % 10 ° i 1010 T [°C 273 tan H 3 i i tan H i " AQ QZ Z\J[YIK[ML NYVT [PM YMZWMK[Q]M IS[Q[\LM& CPM YMNYIK[QVU NVYT\SI QZ VUS` IKK\YI[M NVY IS[Q[\LMZ IJV]M IWWYV_& )(f& =V^MY IS[Q[\LMZ ZPV\SL JM I]VQLML IU`^I` ZQUKM YMNYIK[QVU TI` JMKVTM MYYI[QK IUL ZQUKM [PM IWWIYMU[ LQZR VN [PM TVVU IUL Z\U! IZZ\TMZ IU V]IS ZPIWM KI\ZML J` IU QUKYMIZQUO LQNNMYMUKM QU YMNYIK[QVU NVY \WWMY IUL SV^MY SQTJ& CPQZ LQZ[VY[QVU ^V\SL INNMK[ [PM ZMTQLQITM[MY ^Q[P YMZWMK[ [V [PM YMNMYMUKM JVL` QU I KVTWSQKI[ML ^I`& /%, &. EM KVYYMK[ [PM IS[Q[\LMZ NVY [PM WIYISSI_ QU IS[Q[\LM2 sin P M = sin HP M cos H Mapp ! RMapp " sin PB = sin HP Bcos H Bapp ! RBapp " EM IWWS` [PM IS[Q[\LM KVYYMK[QVUZ IZ NVSSV^Z2 H M = H Mapp ! RMapp PM H B = H Bapp ! RBapp PB CPM KVYYMK[QVU NVY WIYISSI_ QZ UV[ IWWSQML [V [PM IS[Q[\LM VN I NQ_ML Z[IY ;@6 4 (!& '. 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D2 " T 2 = T 1 3h T 3 = T 2 3h D1 & D2 & D3 or D1 % D2 % D3 8 TI` PI]M IU` ]IS\M JM[^MMU 8 ) IUL 8 +& CPMYM T\Z[ UV[ JM I TQUQT\T VY TI_QT\T VN 8 QU [PM [QTM QU[MY]IS GC )$ C+H& CPQZ WYVJSMT LVMZ UV[ VKK\Y ^Q[P I WYVWMYS` KPVZMU JVL` PI]QUO I Z\Q[IJSM YI[M VN KPIUOM VN 8& ?MIY I TQUQT\T VY TI_QT\T VN 8$ 8' C ^V\SL JM ]MY` ZTISS$ IUL [PM VJZMY]I[QVU ^V\SL JM MYYI[QK IU`^I`& 5N[MY NQULQUO C 8$ ^M KIU KISK\SI[M [PM ^I[KP MYYVY$ C& ' T = WT D ! T D C QZ [PM LQNNMYMUKM JM[^MMU V\Y ^I[KP [QTM I[ [PM TVTMU[ VN VJZMY]I[QVU$ EC 8$ IUL [PM [QTM NV\UL J` QU[MYWVSI[QVU$ C8& B\J[YIK[QUO [PM ^I[KP MYYVY NYVT [PM ^I[KP [QTM$ EC$ YMZ\S[Z QU DC& UT = WT !' T /%- -:=> CPM WYVKML\YMZ LMZKYQJML ZV NIY YMNMY [V I ZWPMYQKIS MIY[P& CPM KVYYMK[QVUZ YMX\QYM RUV^SMLOM VN [PM VJZMY]MYcZ SI[Q[\LM$ =I[$ [PM [Y\M IaQT\[P VN [PM TVVU$ 5a >$ IUL [PM [Y\M IaQT\[P VN [PM YMNMYMUKM JVL`$ 5a 6& BQUKM [PM KVYYMK[QVUZ IYM ZTISS$ [PM [PYMM ]IS\MZ LV UV[ UMML [V JM ]MY` IKK\YI[M& 9YYVYZ VN I NM^ LMOYMMZ IYM [VSMYIJSM& @IYISSI_ QU IS[Q[\LM2 CPQZ KVYYMK[QVU QZ IWWSQML [V [PM WIYISSI_ QU IS[Q[\LM IUL QZ \ZML [V KISK\SI[M ;> ^Q[P PQOPMY WYMKQZQVU JMNVYM KSMIYQUO [PM S\UIY LQZ[IUKM& 2 ' PM ( f HP M [sin 2 Lat "cos Az M sin H Mapp ! sin Lat cos H Mapp 1 N QZ [PM NSI[[MUQUO VN [PM MIY[P2 f = 298.257 PM , improved = P M ' PM H M = H mapp ! 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(4;0, )0<, *?45423< %070:,5 $87/4<487; 18: +4;4-454<@ 9UX ZNK VRGTTOTM UL UHYKX\GZOUTY# OZ OY [YKL[R ZU QTU] ZNK ZOSKY J[XOTM ]NOIN G IKXZGOT HUJ_ OY GHU\K ZNK NUXO`UT GY ]KRR GY ZNK ZOSKY UL Y[TXOYK# Y[TYKZ# GTJ Z]OROMNZ% 4 HUJ_ IGT HK GR]G_Y GHU\K ZNK NUXO`UT# GR]G_Y HKRU] ZNK NUXO`UT# UX GHU\K ZNK NUXO`UT J[XOTM G VGXZ UL ZNK JG_# JKVKTJOTM UT ZNK UHYKX\KX'Y RGZOZ[JK GTJ ZNK JKIROTGZOUT UL ZNK HUJ_% 4 HUJ_ OY .4:.=6985,: GR]G_Y GHU\K ZNK IKRKYZOGR NUXO`UT! ]NKT ZNK `KTOZN JOYZGTIK OY YSGRRKX ZNGT /&a GZ ZNK SUSKTZ UL 58?0: SKXOJOGT VGYYGMK# O% K%# ]NKT ZNK HUJ_ OY UT ZNK RU]KX HXGTIN UL ZNK RUIGR SKXOJOGT "&% !- # !% DNOY OY ZNK IGYK OL Lat Dec 90 ° 4 HUJ_ OY IUTZOT[GRR_ HKRU] ZNK IKRKYZOGR NUXO`UT ]NKT ZNK `KTOZN JOYZGTIK OY MXKGZKX ZNGT /&a GZ ZNK OTYZGTZ UL =990: SKXOJOGT VGYYGMK "&% !- $ !% DNOY OY ZNK IGYK OL Lat ! Dec 90 ° 4 IKRKYZOGR HUJ_ HKOTM UT ZNK YGSK NKSOYVNKXK GY ZNK UHYKX\KX OY KOZNKX YUSKZOSKY GHU\K ZNK NUXO`UT UX IOXI[SVURGX% 4 HUJ_ HKOTM UT ZNK UVVUYOZK NKSOYVNKXK OY KOZNKX YUSKZOSKY GHU\K ZNK NUXO`UT UX VKXSGTKTZR_ OT\OYOHRK# H[Z TK\KX IOXI[SVURGX% DNK Y[T VXU\OJKY G MUUJ K^GSVRK UL NU] ZNK \OYOHOROZ_ UL G HUJ_ OY GLLKIZKJ H_ RGZOZ[JK GTJ JKIROTGZOUT% 4Z ZNK ZOSK UL ZNK Y[SSKX YURYZOIK 7KI 2 "()%+a!# ZNK Y[T OY IOXI[SVURGX ZU GT UHYKX\KX HKOTM TUXZN UL ZNK ,:.<4. .4:.50 >GZ 3 ",,%+a!% 4Z ZNK YGSK ZOSK# ZNK Y[T XKSGOTY HKRU] ZNK IKRKYZOGR NUXO`UT GRR JG_ OL ZNK UHYKX\KX OY YU[ZN UL ZNK ,7<,:.<4. .4:.50 >GZ 1 −,,%+a!% 4Z ZNK ZOSKY UL ZNK KW[OTU^KY 7KI 2 &a!# ZNK Y[T OY IOXI[SVURGX UTR_ GZ ZNK VURKY% 4Z ZNK ZOSK UL ZNK ]OTZKX YURYZOIK 7KI 2 $()%+a!# ZNK Y[T OY IOXI[SVURGX YU[ZN UL ZNK GTZGXIZOI IOXIRK GTJ OT\OYOHRK TUXZN UL ZNK GXIZOI IOXIRK% (4;0 ,7/ )0< DNK K\KTZY UL XOYK GTJ YKZ IGT HK [YKJ ZU JKZKXSOTK RGZOZ[JK# RUTMOZ[JK# UX ZOSK% @TK YNU[RJ TUZ K^VKIZ \KX_ GII[XGZK XKY[RZY# NU]K\KX# YOTIK GZSUYVNKXOI XKLXGIZOUT SG_ HK KXXGZOI OL ZNK HUJ_ OY UT UX TKGX ZNK NUXO`UT% .$' DNK MKUSKZXOI XOYK UX YKZ UL G HUJ_ UII[XY ]NKT ZNK IKTZKX UL ZNK HUJ_ VGYYKY ZNXU[MN ZNK IKRKYZOGR NUXO`UT ; 2 &a!% 7[K ZU ZNK OTLR[KTIK UL GZSUYVNKXOI XKLXGIZOUT# GRR HUJOKY K^IKVZ ZNK SUUT GVVKGX GHU\K ZNK \OYOHRK GTJ YKTYOHRK NUXO`UT GZ ZNOY OTYZGTZ% DNK SUUT OY TUZ \OYOHRK GZ ZNK SUSKTZ UL NKX MKUSKZXOI XOYK UX YKZ YOTIK ZNK JKVXKYYOTM KLLKIZ UL ZNK NUXO`UTZGR VGXGRRG^ ∼'a! OY MXKGZKX ZNGT ZNK KRK\GZOTM KLLKIZ UL GZSUYVNKXOI XKLXGIZOUT% DNK GVVXU^OSGZK GVVGXKTZ GRZOZ[JKY XKLKXXOTM ZU ZNK YKTYOHRK NUXO`UT! GZ ZNK SUSKTZ UL ZNK GYZXUTUSOIGR XOYK UX YKZ GXK0 C[T RU]KX ROSH!0 '+' CZGXY0 (/' ARGTKZY0 (/' − ;A FNKT SKGY[XOTM ZNKYK GRZOZ[JKY ]OZN XKLKXKTIK ZU ZNK YKG NUXO`UT# ]K NG\K ZU GJJ ZNK JOV UL NUXO`UT INGVZKX (! ZU ZNK GHU\K \GR[KY% 9UX K^GSVRK# ZNK GRZOZ[JK UL ZNK RU]KX ROSH UL ZNK XOYOTM UX YKZZOTM Y[T OY GVVXU^% (&' OL ZNK NKOMNZ UL K_K OY .S% FK HKMOT ]OZN ZNK ]KRR$QTU]T GRZOZ[JK LUXS[RG INGVZKX *!% sin H = sin Lat "sin Dec cos Lat "cos Dec "cos t = 0 sin Lat "sin Dec cos t = ! cos Lat "cos Dec CUR\OTM ZNK KW[GZOUT LUX ZNK SKXOJOGT GTMRK# Z# ]K MKZ 0 t = arccos #! tan Lat "tan Dec $ DNK KW[GZOUT NGY TU YUR[ZOUT OL ZNK GXM[SKTZ UL ZNK OT\KXYK IUYOTK OY YSGRRKX ZNGT −' UX MXKGZKX ZNGT '% 7[K ZU ZNK GSHOM[OZ_ UL ZNK GXIIUY L[TIZOUT# ZNK KW[GZOUT NGY Z]U YUR[ZOUTY# UTK LUX XOYK GTJ UTK LUX YKZ% 9UX ZNK IGRI[RGZOUTY HKRU]# ]K NG\K ZU UHYKX\K ZNK LURRU]OTM X[RKY0 Lon = !GHA # t :;4 OY ZNK :XKKT]OIN NU[X GTMRK UL ZNK HUJ_ GZ ZNK SUSKTZ UL XOYK UX YKZ% DNK YOMT UL Z NGY ZU HK UHYKX\KJ IGXKL[RR_ YKK GHU\K!% =TU]OTM U[X VUYOZOUT# ]K IGT IGRI[RGZK ZNK ZOSKY UL Y[TXOYK GTJ Y[TYKZ0 Lon [° t [° UT = 12 ! EoT ! # Sunrise , Sunset 15 15 DNK ZOSKY UL Y[TXOYK GTJ Y[TYKZ UHZGOTKJ ]OZN ZNK GHU\K LUXS[RG GXK TUZ W[OZK GII[XGZK YOTIK 7KI GTJ 8UD GXK \GXOGHRK% COTIK ]K JU TUZ QTU] ZNK K^GIZ ZOSK UL XOYK UX YKZ GZ ZNK HKMOTTOTM# ]K NG\K ZU [YK KYZOSGZKJ \GR[KY LUX 7KI GTJ 8UD OTOZOGRR_% DNK ZOSK UL XOYK UX YKZ IGT HK OSVXU\KJ H_ OZKXGZOUT XKVKGZOTM ZNK IGRI[RGZOUTY ]OZN 7KI GTJ 8UD GZ ZNK IGRI[RGZKJ ZOSK UL XOYK UX YKZ!% 9[XZNKX# ZNK ZOSKY ZN[Y IGRI[RGZKJ GXK OTLR[KTIKJ H_ ZNK OXXKM[RGXOZOKY UL GZSUYVNKXOI XKLXGIZOUT TKGX ZNK NUXO`UT% DNKXKLUXK# G ZOSK KXXUX UL ±( SOT[ZKY OY TUZ [T[Y[GR% 4IIUXJOTMR_# ]K IGT IGRI[RGZK U[X RUTMOZ[JK LXUS ZNK ZOSK UL Y[TXOYK UX Y[TYKZ OL ]K QTU] U[X RGZOZ[JK0 15 12 Lon [° = "# ! GMT Sunrise , Sunset ! EoT $ # t [° .$( 4MGOT# ZNOY OY TUZ G \KX_ VXKIOYK SKZNUJ# GTJ GT KXXUX UL YK\KXGR GXISOT[ZKY OT RUTMOZ[JK OY TUZ [TROQKR_% =TU]OTM U[X RUTMOZ[JK# ]K GXK GHRK ZU JKZKXSOTK U[X GVVXU^OSGZK RGZOZ[JK LXUS ZNK ZOSK UL Y[TXOYK UX Y[TYKZ0 15 12 t [° = Lon [° ! "# ! GMT Sunrise, Sunset ! EoT $ cos t Lat = arctan ! # tan Dec $ &7 7,>42,<487, :4;0 ,7/ ;0< ,:0 /01470/ ,; <30 68607<; ?307 <30 =990: 546- 81 , -8/@ 4; 87 <30 >4;4-50 38:4A87 DNKYK K\KTZY IGT HK UHYKX\KJ ]OZNU[Z G YK^ZGTZ% ?U]# ]K NG\K ZU ZGQK OTZU GIIU[TZ ZNK KLLKIZY UL XKLXGIZOUT# NUXO`UTZGR VGXGRRG^# JOV# GTJ YKSOJOGSKZKX% DNKYK W[GTZOZOKY JKZKXSOTK ZNK GRZOZ[JK ;U! UL G HUJ_ ]OZN XKYVKIZ ZU ZNK IKRKYZOGR NUXO`UT GZ ZNK OTYZGTZ UL ZNK \OYOHRK XOYK UX YKZ% sin Ho ! sin Lat "sin Dec t = arccos cos Lat "cos Dec Ho = HP ! SD ! RH ! Dip 5_ JKLOTOZOUT# ZNK YZGTJGXJ XKLXGIZOUT LUX G HUJ_ HKOTM UT ZNK YKTYOHRK NUXO`UT# B;# OY )*' OT XKGROZ_ OZ OY Y[HPKIZ ZU XGTJUS \GXOGZOUTY!% FNKT UHYKX\OTM ZNK [VVKX ROSH UL ZNK Y[T# ]K MKZ0 Ho = 0.15 ' ! 16 ' ! 34 ' ! Dip %!50 ' ! Dip ;U OY TKMGZO\K% #A46=<3 ,7/ #6954<=/0 DNK G`OS[ZN GTMRK UL G XOYOTM UX YKZZOTM HUJ_ OY IGRI[RGZKJ ]OZN ZNK G`OS[ZN LUXS[RG YKK INGVZKX *!0 sin Dec ! sin H "sin Lat Az = arccos cos H "cos Lat FOZN ;2&# ]K MKZ0 sin Dec Az = arccos cos Lat 4` OY "/&a XOYK! GTJ −/&a YKZ! OL ZNK JKIROTGZOUT UL ZNK HUJ_ OY `KXU# XKMGXJRKYY UL ZNK UHYKX\KX'Y RGZOZ[JK% 4IIUXJOTMR_# ZNK Y[T XOYKY OT ZNK KGYZ GTJ YKZY OT ZNK ]KYZ GZ ZNK ZOSKY UL ZNK KW[OTU^KY MKUSKZXOI XOYK GTJ YKZ!% − FOZN ; IKTZKX 2 +&' [VVKX ROSH UL ZNK Y[T UT ZNK YKTYOHRK NUXO`UT!# ]K NG\K0 sin Dec 0.0145 "sin Lat Az = arccos 0.9999 "cos Lat DNK ZX[K G`OS[ZN UL ZNK XOYOTM UX YKZZOTM HUJ_ OY0 if 0 Az = Az t & N !360 °! Az if t 0 " DNK G`OS[ZN UL G HUJ_ GZ ZNK SUSKTZ UL XOYK UX YKZ IGT HK [YKJ ZU LOTJ ZNK SGMTKZOI JKIROTGZOUT GZ ZNK UHYKX\KX'Y VUYOZOUT IUSVGXK ]OZN INGVZKX ')!% .$) DNK NUXO`UTZGR GTM[RGX JOYZGTIK UL G XOYOTM UX YKZZOTM HUJ_ LXUS ZNK KGYZ XOYOTM! UX ]KYZ YKZZOTM! VUOTZ UT ZNK NUXO`UT OY IGRRKJ ,6954<=/0 GTJ IGT HK IGRI[RGZKJ LXUS ZNK G`OS[ZN% 4T GSVROZ[JK UL 8*+a?# LUX OTYZGTIK# SKGTY ZNGZ ZNK HUJ_ XOYKY *+a TUXZN UL ZNK KGYZ VUOTZ UT ZNK NUXO`UT% *?45423< 4Z YKG# Z]OROMNZ OY OSVUXZGTZ LUX ZNK UHYKX\GZOUT UL YZGXY GTJ VRGTKZY YOTIK OZ OY ZNK UTR_ ZOSK ]NKT ZNKYK HUJOKY ,7/ ZNK NUXO`UT GXK \OYOHRK% 5_ JKLOTOZOUT# ZNKXK GXK ZNXKK QOTJY UL Z]OROMNZ% DNK GRZOZ[JK# ;# XKLKXY ZU ZNK IKTZKX UL ZNK Y[T GTJ ZNK IKRKYZOGR NUXO`UT GTJ SGXQY ZNK HKMOTTOTM SUXTOTM! GTJ ZNK KTJ K\KTOTM! UL ZNK XKYVKIZO\K Z]OROMNZ% $4>45 ',=<4.,5 #;<:87864.,5 DNK SKXOJOGT GTMRK LUX ZNK Y[T GZ −,a GRZOZ[JK IKTZKX! 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( %; * ANK LUXS[RG LUX ZNK VGXGRRG^ OT G`OS[ZN OY GRYU GII[XGZK ZU G LXGIZOUT UL GT GXIYKIUTJ% :Z HKIUSKY RKYY GII[XGZK GY ZNK GRZOZ[JK GVVXUGINKY 0'c% >HYKX\OTM HUJOKY ]OZN Y[IN GRZOZ[JKY# NU]K\KX# OY JOLLOI[RZ GTJ [Y[GRR_ G\UOJKJ% -97=@=CA ;5B8=4! 7UX K\KT SUXK GII[XGZK XKY[RZY# ]K [YK ZNK B=>=35 ANK ZUVUIKTZXOI KW[GZUXOGR IUUXJOTGZKY UL G IKRKYZOGR HUJ_ GXK UHZGOTKJ LXUS ZNK MKUIKTZXOI UTKY ZNXU[MN IUUXJOTGZK ZXGTYLUXSGZOUT% ANK MO\KT W[GTZOZOKY GXK1 8KUMXGVNOI RGZOZ[JK UL ZNK UHYKX\KX ;GZ 8KUIKTZXOI SKXOJOGT GTMRK Z 8KUIKTZXOI JKIROTGZOUT 5KI 6W[GZUXOGR NUXO`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r Lat ' = arctan p !tan Lat = arctan [ 1 f 2 !tan Lat [ re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sin A1 sin A2 sin A3 = = sin s1 sin s2 sin s3 '.B :3 0:>692> 3:= >612># cos s1 = cos s2 cos s3 sin s2 sin s3cos A1 cos s2 = cos s1 cos s3 sin s1 sin s3 cos A2 cos s3 = cos s1 cos s2 sin s1 sin s2cos A3 '.B :3 0:>692> 3:= .9472># cos A1 = cos A2cos A3 sin A2 sin A3 cos s1 cos A2 = cos A1cos A3 sin A1sin A3 cos s2 cos A3 = cos A1cos A2 sin A1sin A2 cos s3 &%#& '.B :3 >692> .91 0:>692># LBMK CG>C=;L?K LB? IE;G? H@ LB? K?GKC