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ANHANG Zahlenbeispiele Allgemeine Hinweise 1. in Den Folgenden ANHANG I. TEIL Zahlenbeispiele Allgemeine Hinweise 1. In den folgenden Zahlenbeispielen wurde in mehreren Fallen unterstellt, da/3 die Zeitsignale der astronomischen Weltzeit UT 0 entsprechen, da die Be­ obachtungen teilweise vor der Beriicksichtigung der Unregelmi:i/3igkeiten der Erddrehung bei der Zeitfestlegung gemacht wurden, und/oder die Signalkorrek­ tionen bei der Auswertung nicht bekannt waren. Die Laufzeit der Signale vom Sender bis zum Empfi:inger wurde in diesen Fallen nicht beriicksichtigt. Auf eine nachtragliche Beriicksichtigung der Signalkorrektionen und der Laufzeit wurde verzichtet, da die Absolutwerte der Zeit-, Langen- und Azimutbestim­ mungen fiir das Handbuch eine untergeordnete Bedeutung haben, und Breiten­ bestimmungen dadurch nicht beeinflu/3t werden. Die strenge Ermittlung der W eltzeiten UT 2 und UT 0 durch Anschlu/3 der Beobachtungsuhr an Zeitsignale wird in Zahlenbeispiel 1 gezeigt. 2. Abkiirzungen: App. Pl. 64 = Apparent Places of Fundamental Stars 1964, AE 64 = The Astronomical Ephemeris 1964, Bull. Hor. = Bulletin Horaire, Serie J, BIR, Circulaire D = Bureau International de l'Heure, Circulaire D. 3. Die Herkunft der in den Zahlenbeispielen angegebenen Formeln wird durch die Nummern des Paragraphen und der Formel angegeben, z.B. § 14(3). 1. Ermittlung der Weltzeiten UT2 und UTO durch AnschluB der Beobachtungs­ uhr an Zeitsignale Die am 10. Juli 1968 auf der astronomischen Beobachtungsstation der Uni­ versiti:it Stuttgart (<pr:::: 48° 4 7', AE r:::; 9° 13') mit einem geni:ihert auf Weltzeit ein­ regulierten Druckchronographen registrierte Uhrzeit U = ist in die entsprechenden Weltzeiten U T2 und U TO umzurechnen. Die kurz zuvor durch den Anschlu/3 an den Sender DIZ Nauen, der die vom Geodi:itischen In- 702 ZAHLENBEISPIELE stitut Potsdam ermittelten Dauerzeitsignale ausstrahlt, ohne Beriicksichtigung der Laufzeit der elektromagnetischen Wellen bestimmte Uhrkorrektion betragt u = + OU49. Der Uhrgang ist vernachlassigbar klein. Die Entfernung Nauen­ Stuttgart betragt 500km. Da der Sender im Kurzwellenbereich (Wellenlange 66,3 m) arbeitet, ist nach § 38 (8) fiir die Ausbreitungsgeschwindigkeit der elek­ tromagnetischen Wellen v = 280000 km/sec zu setzen. Unter Beriicksichtigung von § 38(4), (6), (9) und § 14(11), (12) gilt. UTO = Sollzeit Signal+ E +UL= U + u + E +UL, UL= s/v, UT2 UTO + (UT2 - UTO), UTO UT 0 + (UT 1 - UT 0) - LI)., 1• § 14(13): LI ).s =(UT 1 - U TO)•= 1511 (x ·sin Aw -y ·cos Aw)· tg<p. 1. Berechnung von UT 0 u = U= + 0,149 E = 0,0004 (BIH, Circulaire D 22) UL= + 0,0018 UTO = 2. Berechnung von UT 2 UTO = (UT 2 - UT 0) = + 0,0280 (BIH, Circulaire D 22) UT2 = 3. Berechnung von LI). x = + o:on, y = + 0:181, (BIH, Circulaire D 22), Aw= -9°13', sinlw= -0,160, cosAw= +0,987; <p=48°47', tg<p= 1,142, 1• LI;. = - (- o:on · 0,160- 0:181·o,987).1,142 = - 15" 4. Berechnung von UT 0 UTO= (UT1 - UTO) = + 0,0147 (BIH, Circulaire D 22) - LI).= + 0,0145 UTO = Anmerkung: Ist die Reduktion der Zeitsignale auf UT2 bzw. UTO bei der Auswertung der Beobachtungen noch nicht bekannt, so beriicksichtigt man vor­ laufig nur die Laufzeit UL. Nach dem Bekanntwerden der Signalkorrektion wer­ den dann die aus den Beobachtungen abgeleiteten vorlaufigen Werte der Lange bzw. Zeit oder des Azimuts entsprechend verbessert. Eine Verbesserung der Breite ist in der Regel nicht erforderlich, da bei den meisten Verfahren zur Brei­ tenbestimmung ein kleiner Zeitfehler ohne Einflul3 ist. ZAHLENBEISPIELE 703 Es sei auf die Arunerkung in§ 111 iiber den vom BIH 1959-67 verwendeten mittleren Pol verwiesen. Um UTO im neuen System 1900-05, also bezogen auf den internationalen konventionellen Ursprung, zu erhalten, geniigt es nicht, bei der Berechnung von L1 A. die Polkoordinaten des IPMS zu verwenden oder die Polkoordinaten des BIH um L1x = + L1y = + zu verbessern; auch die UT 2 def. mul3 auf das neue System 1900-05 reduziert werden. Die dazu not­ wendigen Angaben sind im ,,Rapport annuel pour 1967" des BIH auf Seite 38 zu finden. 2. Umrechnung von Weltzeit UTO in Sternzeit Der Zeitpunkt 10. Juli 1968 ist in Sternzeit, bezogen auf den Meridian AE = 9° 12' umzurechnen. § 16(18) und (13): 6)h = B8,ar + UTOh + {} UTOh ± + L1lp · cose. § 16(10) : {} = 0,0027379093. § 13(7): L1 lp • cos e = N' + N" = Gleichung der Aquinoktien. N' = langperiodischer Anteil, N" = kurzperiodischer Anteil. L1 lp · cos e ist in AE, N' und N" in App. Pl. vertafelt. B8, Gr= 19h ) N'= 0,200 App. Pl. 68, S. 482 N"= + 0,016 U TOh = 22 16 28,014 {} · UTO = + 3 39,548 Anhang Tafel 5 + = + 36 51,025 Anhang Tafel 3 6> = 3. Umrechnung von Sternzeit in Weltzeit Aus astronomischen Beobachtungen wurde am 10. Juli 1968 fiir den Beobachtungsort rp 48°47', AE = = 36ID5H025 die Sternzeit 6> = ermittelt. Diese Zeit ist in die Weltzeiten UTO und UT2 umzurechnen. 1. Berechnung von UT 0 § 16(20): ear= 6>-L1lp. cose =f § 13 (7): L1 lp · cos e = N' + N" = Gleichung der Aquinoktien. N' = langperiodischer Anteil, N" = kurzperiodischer Anteil. L1 lp • cos e ist in A E, N' und N" in App. Pl. vertafelt. -h - iii -h -h iii §16(19): UTOli= (Bar- B8,ar) · ih -'f} ·(Bar- Bo, Gr)· th" § 16 (9): 'f/ = 0,0027304336. 704 ZAHLENBEISPIELE Da zur Interpolation von LI "P • coss = N' + N" UT genahert bekannt sein mul3, rechnen wir besser nach folgender Gleichung E - iii E -h iii UTOli = (@h =f J.hw- ih -11. (@h =f A.hw- <">o,Gr). ih iii iii - LI tp • cos s · - 11 · LI "P • cos s · - . 1h + 1h 1st @Gr bzw. (@ =f @o,Gr, so sind 24h zu addieren. e = - AE = - 36 51,025 e - AE = 17h 32m @0 , Gr= 19 11 55,i64 (AE 68, S. i4) - iii (@- AE- <">0, Gr)· ih = -11(<">-AE- <">o,Gr) 3 39,547 (Anhang Tafel 6) UT Oli -Ll'l/)·COSS= + 0,184 (AE 68, S. 14) + 11 • Ll'I/) ·COSS= - 0,0Qi (Anhang Tafel 6) UTOli = 2. Berechnung von UT 2 § i4(11), (12): UT2 = UTO+ LI l+LITs = UTO+LI A.+ (UT2- UT1) = UTO+Lll+ (UT2- UTO)-(UT1- UTO). i• § 14(13): LI J.s = (UT1 - U TO)•= 15,,(x · sinA.w-y · cosA.w) · tg<p. x = + 0:073, y = + o:isi (BIR, Circulaire D 22). lw = -9°13', sinlw = -O,i60, cosA.w = 0,987; <p = 48°47', tg<p = 1,142. 1• LI J. = -(- 0:073 · 0,160 - 0:1s1·0,987)·1,142 = - 15" UTO = LI A.= 0,0145 (UT2-UT0)= + 0,0280} . _ (UTi _ UTO) = O,Oi 47 (BIR, C1rculaire D 22) UT2 = 4. Ermittlung der Rektaszension und Deklination der Sonne Gesucht wird die scheinbare Rektaszension und Deklination der Sonne am 5. Juni 1965 10h25mo1• Weltzeit UT2. Wir verwandeln zunachst UT 2 in Ephemeridenzeit ET durch Addition von LIT, das wir aus den Astronomical Ephemeris 1965, Seite VII, entnehmen. Wir erhalten ZAHLENBEISPIELE 705 Dail die Zeitdifferenz LI T nur auf volle Sekunden angegeben werden kann, - der genaue Wert kann erst nachtraglich berechnet werden - ist fiir den vor­ liegenden Zweck nicht storend, da sich die Rektaszension und Deklination von Sonne, Mond und Planeten nur langsam andern. In den Astronomical Ephemeris 1965 finden wir ferner auf Seite 25 folgende Werte fiir ex und lJ jeweils fiir Oh ET vertafelt: ex lJl lJ2 lJ3 4. Juni + 5. Juni 4 51 05,58 + + 247,13 6. Juni 4 55 12,71 + 0,29 + 247,42 7. Juni 4 59 20,13 + 0,61 lJ lJl lJ2 lJ3 4. Juni + 22°231 01:1 + 418:0 5. Juni + 22 29 59,1 -23:6 + 394,4 6. Juni + 22 36 33,5 -24,0 - 0:4 + 370,4 7. Juni + 22 42 43,9 -47,6 Zur Berechnung von p = (10h25m368 )/24h verwenden wir die Tafel zur Ver­ wandlung von Stunden, Minuten und Sekunden in Dezimalteile des Tages (An­ hang Tafel 4). Dabei ist darauf zu achten, daLl p bei der Interpolation von ex (lJ) auf mindestens 5(4) Stellen ermittelt wird, da die ersten Differenzen von ex (lJ) fiinfstellige (vierstellige) Zahlen sind und einem Fehler dp = 1·10-5 (1 · 10-4) der Interpolationsfehler dex = (dlJ = o:04) entspricht. Wir erhalten 10h25m = 0,434028 368 = 0,000 417 p = 0,434445. exo = + p ·lJ,12 = + 0,434445 · 247U3 = + + B2 · (lJ3 + lJf) = - 0,0614 · lJo = + 22° 29' 59:1 + p. lJ,,. = + 0,434445. 394:4 = + 2'51:35 + B2 • (lJ3 + lJf) = - 0,0614 · (- 47:6) = + 2:92 lJ = + 22° 32' 53:4 Dabei wurde B 2 aus Anhang Tafel 7 entnommen. 5. Ermittlung der Rektaszension und Deklination eines Fixsterns Gesucht wird die scheinbare Rektaszension und Deklination von lJ Persei am 18. Marz 1965 fiir die mittlere Sternzeit e = bezogen auf den Ort .ii.= 9°14' 43n Ost, cp = 48°47'15n Nord. Nach den App. Pl. 65, S. 58, ist J.E.K. Ila 45 706 ZAHLENBEISPIELE 151 Marz 2.7 a= 3h40m24U55 - 244 t2.7 +27 -2t7 +to 22.7 + 37 - t80 April 1.6 + 64 151 Marz 2.7 15 = + - 88 t2.7 46,03 -24 -112 +t 22.7 44,9t -23 - t35 April 1.6 43,56 -47 Berechnung von UT fiir den Interpolationszeitpunkt D: - - h - t Ii -Ii E -h t Ii U (@Gr - @8, Gr)· ((0 =f A.hw)- 00,Gr) · th· @h - (Anh. Tafel 3) R> t 3h 50m Ei8,Gr (App. Pl. 65. Tafel II) R> 11h4tm U Tli R> 21i 09fii 04090 D R> Marz Berechnung von UT fiir obere Kulmination Greenwich am t2. Marz: iii § t9(7) UTli (och - @8 • Gr)·-.th 3h40m 11ht8m D0 Marz t2,68. Berechnung des Interpolationsfaktors p und des Besselkoeffizienten Ba: § t9(8):p=0,t003 (D-D0 ). p = 0,1003 (t8,09 - t2,68) = O,t003. 5,41 = 0,543. Ba= - 0,0620 (Anh.
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