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Astronomia
Formule e Tabelle
Rodolfo Baggio
Milano, ottobre 2000 ultima revisione: dicembre 2001
2
Sommario
INTRODUZIONE 4
TEMPI 5
Definizioni 5 Ore / gradi 7 Tempo siderale 7 Sorgere e tramonto 7
CRONOLOGIA 8
Calendario perpetuo 8 Transizione dal calendario Giuliano a quello Gregoriano 9 Calcoli sul calendario 10
AA (numero dell'anno) 10 Anno Bisestile 10 Numero solare e Lettera Domenicale 10 Epatta e Numero Aureo 11 Numero del giorno nell'anno 11
Data di Pasqua 13 Giorno Giuliano (JD) 13
Calcolo di JD 14 Giorno Giuliano Modificato 14 Epoca standard J2000.0 14 Numero di giorni dall'epoca J2000.0 14 JD approssimato 14 Giorno della settimana 14 Tabelle per il calcolo di JD 15 Astrodate 17
Cicli periodici 17 Altri calendari 18
COORDINATE CELESTI 19
Trasformazione di coordinate 19 Orizzontali ed Equatoriali 19 Equatoriali e Eclittiche 19 Altezza e Azimuth 19 Altezza sull'orizzonte al meridiano 20 Azimuth dei punti di sorgere e tramonto 20 Declinazione di un oggetto sull'eclittica 20 Ora approssimata di alba / tramonto 20 Correzione all'altezza per rifrazione 20 Allineamenti 20 Potere risolutivo dell'occhio 21 Semiarco Diurno 22
PRECESSIONE 24
Precessione approssimata 24 Coefficienti per il calcolo della precessione 25
3
SOLE, LUNA E PIANETI 27
Posizioni di Sole, Luna e Pianeti 27 Massima declinazione della Luna 28 Età della Luna 28
Posizione del Sole 28 Declinazione approssimata del Sole 29 Massima declinazione del Sole 29 Equazione del Tempo 29 Posizioni approssimate del Sole 29 Altezza del Sole durante il giorno 30 Apogeo (Auge) Solare 30
DELTA T 31
COORDINATE DELLE STELLE PIÙ BRILLANTI (J2000.0) 33
TRIGONOMETRIA PIANA 34
Definizioni 34 Identità trigonometriche 34 Elementi di un triangolo 34 Funzioni di angoli opposti, complementari, supplementari 35 Valori principali 35 Calcolo dell'arcotangente 35
TRIGONOMETRIA SFERICA 36
Sfera trigonometrica 36 Principali relazioni trigonometriche 38 Triangoli sferici rettangoli 39
SIMBOLI 40
COSTELLAZIONI 43
Costellazioni obsolete 45
TAVOLE TRIGONOMETRICHE 46
BIBLIOGRAFIA 49
CARTE E MAPPE 50
CATALOGO STORICO 58
CATALOGO DI YALE 83
COPYRIGHT 89
4
Introduzione Questo lavoro è dedicato a tutti coloro i quali hanno interesse o necessità nell'utilizzo dell'astronomia per ricerche storiche. Un lavoro come questo può sembrare un controsenso in un'epoca in cui la diffusione di sistemi di calcolo è così elevata e in cui uno qualunque delle svariate centinaia di software disponibili può fornire tutte le informazioni richieste con grande precisione e a costi molto ridotti se non inesistenti. Ma non sempre il computer è a portata di mano, spesso la familiarità con i programmi non è tanta, o, semplicemente, non si ha il tempo o la voglia di accendere il computer, far partire il programma, stampare i risultati. O magari ci si trova in un posto dove non è possibile usare un computer. E poi molti dati non sono sempre facilmente ottenibili da tutti i programmi, costruiti più per l'osservatore amatoriale di oggi che non per chi fa ricerche storiche. A volte è più rapido usare una formula approssimata, fare qualche calcolo a mente o con una calcolatrice tascabile o cercare un valore in una tabella già calcolata. Spesso, infatti, quello che serve è un modo rapido per una prima verifica, i conti precisi si faranno poi. In questo volume troverete formule approssimate per il calcolo di tempi e posizioni dei corpi celesti (stelle e pianeti), per calcoli riguardanti le date, le tabelle con i coefficienti per il calcolo della precessione e tavole trigonometriche e aritmetiche. Infine il catalogo con le posizioni delle stelle più brillanti calcolate riducendole per la precessione ogni cento anni fino al 4713 a.C.
5
Tempi
Definizioni
Anno Durata Descrizione Anno tropico 365d05h48m45.98s Intervallo tra due ritorni del Sole allo stesso punto equinoziale
(equinozio di primavera). Anno civile 365d o 366d Anno del calendario (giuliano o gregoriano) che mantiene una
durata media molto prossima a quella dell'anno tropico con il meccanismo degli anni bisestili.
Anno sidereo o siderale
365d06h09m09.54s Tempo impiegato dalla Terra a compiere una rivoluzione intorno al Sole, ovvero l'intervallo tra due ritorni del Sole allo stesso punto rispetto alle stelle fisse. La differenza tra quest'anno e quello tropico è nota come precessione degli equinozi.
Anno besseliano 365d05h48m45.98s Intervallo tra due ritorni del Sole a longitudine = 280° Questo anno dunque non inizia alle ore 0h del 1 Gennaio, ma qualche ora dopo o prima a seconda degli anni.
Anno anomalistico 365d06h13m53.01s Intervallo tra due ritorni del Sole al perigeo (o della Terra al perielio).
Anno del'eclissi 346d14h53m46.33s Intervallo tra due ritorni del Sole su un nodo dell'orbita.
Mese Durata Descrizione Mese sidereo o siderale
27d 7h 43m 11.5s Intervallo di tempo compreso tra due congiunzioni successive della Luna con la stessa stella
Mese tropico 27d 7h 43m 4.7s Intervallo di tempo compreso tra due passaggi successi della Luna attraverso il cerchio orario dell'equinozio di primavera.
Mese sinodico o lunazione
29d 12h 44m 2.9s Intervallo di tempo compreso tre due fasi lunari uguali e successive
Mese anomalistico 27d 13h 18m 33.2s Intervallo di tempo compreso tra due passaggi consecutivi della Luna al perigeo
Mese draconico o draconitico
27d 5h 5m 35.8s Intervallo di tempo compreso tra due passaggi successivi della Luna per il nodo ascendente della sua orbita
Giorno Durata Descrizione Giorno sidereo o siderale
23h 56m 04.0905s
= 86400s Intervallo di tempo compreso tra due passaggi successivi del punto vernale (punto di longitudine celeste = 0°) al meridiano.
Giorno solare medio 86164.099s
Giorno delle effemeridi
Secondo delle effemeridi: definito in in base all'anno tropico 1900: Anno 1900 = 31 556 925.9747 s
Altre definizioni:
• Ora solare locale: definita in base al moto del Sole, al passaggio al meridiano locale sono le ore 12 (detta anche ora solare apparente).
• Ora civile: definita come ora solare media ai meridiani centrali dei fusi, uno ogni 15°; il
"Sole medio" ha velocità angolare costante. La differenza fra Sole medio e Sole vero è data dall'equazione del tempo.
6
• UT (Tempo Universale): ora civile del meridiano 0° (Greenwich, detto anche GMT, Z). UT1 è UT corretto per i moti del polo geografico, UT2 è UT1 con correzioni per il rallentamento del moto terrestre.
• TAI (Tempo Atomico Internazionale): Media dei tempi degli orologi atomici di riferimento.
Un secondo (SI) = 9 192 631 770 cicli della radiazione corrispondente alla transizione fra due livelli iperfini dello stato base del Cesio 133. 0:00:00 TAI = 0:00:00 UT2 al 1/1/1958
• UTC (Tempo Universale Coordinato): definito in base a TAI , è UT con correzioni per i
moti della Terra, sta entro 0.9 s da UT1. Differisce da TAI per un numero intero di secondi. Es.: 2000.01.01 00:00:00 UTC = 2000.01.01 00:00:34 TAI
• ET (Tempo delle Effemeridi): un secondo = 1 / 31 556 925.9747 della durata dell'anno
tropico 1900. Il Sole medio delle effemeridi ha ascensione retta data da: AR=18h38m45.836s + 8,640,184.542s · TE + 0.0929s · TE2 dove TE è il numero di secoli Giuliani (36525 giorni) dal 1900 0.5 gennaio. ET - UT = 24.349s + 72.318s · TE + 29.950s · TE2
• TDT (Tempo Dinamico Terrestre): 86400 secondi SI sul geoide terrestre. TDT = TAI +
32.184 secondi. Poi: UT-UTC = TAI-UTC - (TDT-UT) + 32.184s • Delta T: differenza fra il tempo del moto di rotazione terrestre (UT1) e TDT.
7
Ore / gradi ore = gradi / 15 gradi = ore · 15
Tempo siderale
GST =(UT · 1.002737909) + (6.697374558 + 2400.051336·T + 0.000025862·T2) LST = GST - L/15
dove:
UT = ora di Greenwich GST = Tempo siderale a Greenwich (ore) LST = Tempo siderale locale (ore) JD = giorno giuliano T = ( JD - 2451545.0 ) / 36525.0 L = Longitudine geografica: est -, ovest +
Sorgere e tramonto Calcolare LST0, tempo siderale locale a 0h UT. L'ora del Meridiano MT è:
MT = α - LST0 α (ascensione retta dell'oggetto) in gradi. Se MT < 0 aggiungere 360°. Se l'oggetto è una stella:
MT = MT · 365.2422/366.2422 = MT · 0.9972696
Calcolare l'angolo orario del sorgere/tramonto:
δ⋅ϕδ⋅ϕ−
=coscos
sinsinsinhHAcos 0
0
dove h0 è l'altitudine al sorgere/tramonto. Per un orizzonte matematico h = 0 quindi:
cos HA0 = - tan ϕ · tan δ Per tener conto della rifrazione: h0 = -35/60 = 0.583 (-35 min d'arco), per il bordo superiore del Sole h0 = -50/60 = 0.833. Poi, se l'oggetto è una stella:
HA0 = HA0 · 365.2422/366.2422 = HA0 · 0.9972696 Infine:
Sorgere = MT - HA0 Tramonto = MT + HA0
Per avere il tempo in ore convertire dividendo per 15.
8
Cronologia
Calendario perpetuo Anno Secolo
Giuliano Gregoriano 0 100 200 300 400 500 600 1500* * 1600 1700 1800 1900 700 800 900 1000 1100 1200 1300 2000 2100 2200 2300 1400 1500* 0 DC ED FE GF AG BA CB ... BA C E G 1 29 57 85 B C D E F G A F G B D F 2 30 58 86 A B C D E F G E F A C E 3 31 59 87 G A B C D E F D E G B D 4 32 60 88 FE GF AG BA CB DC ED CB DC FE AG CB 5 33 61 89 D E F G A B C A B D F A 6 34 62 90 C D E F G A B G A C E G 7 35 63 91 B C D E F G A F G B D F 8 36 64 92 AG BA CB DC ED FE GF ED FE AG CB ED 9 37 65 93 F G A B C D E C D F A C 10 38 66 94 E F G A B C D B C E G B 11 39 67 95 D E F G A B C A B D F A 12 40 68 96 CB DC ED FE GF AG BA GF AG CB ED GF 13 41 69 97 A B C D E F G E F A C E 14 42 70 98 G A B C D E F D E G B D 15 43 71 99 F G A B C D E C D F A C 16 44 72 ED FE GF AG BA CB DC ... CB ED GF BA 17 45 73 C D E F G A B ... A C E G 18 46 74 B C D E F G A ... G B D F 19 47 75 A B C D E F G ... F A C E 20 48 76 GF AG BA CB DC ED FE ... ED GF BA DC 21 49 77 E F G A B C D ... C E G B 22 50 78 D E F G A B C ... B D F A 23 51 79 C D E F G A B ... A C E G 24 52 80 BA CB DC ED FE GF AG ... GF BA DC FE 25 53 81 G A B C D E F ... E G B D 26 54 82 F G A B C D E C D F A C 27 55 83 E F G A B C D B C E G B 28 56 84 DC ED FE GF AG BA CB AG BA DC FE AG
Mese Lettera domenicale
Gen, Ott A B C D E F G Feb, Mar, Nov D E F G A B C Apr, Lug G A B C D E F Mag B C D E F G A Giu E F G A B C D Ago C D E F G A B Set, Dic F G A B C D E
Giorni 1 8 15 22 29 Dom Sab Ven Gio Mer Mar Lun 2 9 16 23 30 Lun Dom Sab Ven Gio Mer Mar 3 10 17 24 31 Mar Lun Dom Sab Ven Gio Mer 4 11 18 25 Mer Mar Lun Dom Sab Ven Gio 5 12 19 26 Gio Mer Mar Lun Dom Sab Ven 6 13 20 27 Ven Gio Mer Mar Lun Dom Sab 7 14 21 28 Sab Ven Gio Mer Mar Lun Dom
9
Leggere la lettera domenicale all'incrocio fra le colonne che contengono l'anno e il secolo. Se ci sono due lettere (anno bisestile) la prima vale per gennaio e febbraio, la seconda per i mesi da marzo a dicembre. Nel settore dei mesi si cerca in quale colonna la lettera appare sulla stessa linea orizzontale del mese considerato. Infine, nella stessa colonna verticale, nel settore dei giorni, si trova il giorno della settimana che appare all'incrocio con la riga in cui sta il giorno del mese considerato.
Transizione dal calendario Giuliano a quello Gregoriano 1582, 15 ottobre Italia, Spagna, Portogallo e parti cattoliche della
Polonia: venerdì 15 ottobre segue giovedì 4 ottobre.
1582, 20 dicembre Francia e Lorena: lunedì 20 dicembre segue domenica 9 dicembre.
1583, 1 gennaio Paesi Bassi (Brabant, Flanders, Hennegan e some province del sud): sabato 1 gennaio segue venerdì 21 dicembre.
1583, 16 ottobre Regioni cattoliche della Germania: domenica 16 ottobre segue sabato 5 ottobre.
1583 e 1584 Austria Tirolo: domenica 16 ottobre segue sabato 5
ottobre. Carinzia (Kärnten) e Stiria (Steiermark): domenica 25 dicembre segue sabato 14 dicembre.
1584 Boemia, Moravia (Mähren) e Lusazia (Lausitz): martedì 17 gennaio segue lunedì 6 gennaio. Slesia (Schlesien): lunedì 23 gennaio segue domenica 12 gennaio.
1584 Regioni cattoliche della Svizzera: domenica 22 gennaio segue sabato 11 gennaio.
1587, 1 novembre Ungheria: domenica 1 novembre segue sabato 21 ottobre.
1590, 25 dicembre Transilvania (Siebenbürgen): martedì 25 dicembre segue lunedì 14 dicembre.
1610, 2 settembre Ducato di Prussia: giovedì 2 settembre segue mercoledì 22 agosto.
1700, 1 marzo Germania protestante, Danimarca e Norvegia: lunedì 1 marzo segue domenica 18 febbraio.
1700 Paesi Bassi Gelderland: lunedì 12 luglio segue domenica 30
giugno. Utrecht e Overijssel: domenica 12 dicembre segue
sabato 30 novembre. Friesland e Groningen: mercoledì 12 gennaio
segue martedì 31 dicembre. 1701, 12 gennaio Svizzera (cantoni di Zurich, Bern, Basel,
Schaffhausen, Gent, Mühlhausen e Biel): mercoledì 12 gennaio segue martedì 31 dicembre.
1752, 14 settembre Inghilterra e Scozia con le colonie: giovedì 14 settembre segue mercoledì 2 settembre.
1753, 1 marzo Svezia: fra il 1700 e il 1712 adottato un calendario né giuliano né gregoriano. Dal 1712 ripristinato il
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calendario giuliano. Nel 1753 giovedì 1 marzo segue mercoledì 17 febbraio.
1873 Giappone. 1912 China. 1914, 1 gennaio Turchia: il calendario precedente era islamico 1918, 14 febbraio Russia: giovedì 14 febbraio segue mercoledì 31
gennaio. 1919, 14 aprile Romania: lunedì 14 aprile segue domenica 31
marzo 1923, 1 marzo Grecia: giovedì 1 marzo segue mercoledì 15
febbraio.
Numero di giorni di ritardo del calendario giuliano rispetto a quello gregoriano: 1582 - 1699 10 giorni 1700 - 1799 11 giorni 1800 - 1899 12 giorni 1900 - 2099 13 giorni 2100 - 2199 14 giorni
Calcoli sul calendario
AA (numero dell'anno) se anno d.C. AA = AA se anno a.C. AA = -(AA - 1)
Anno Bisestile Calendario Giuliano ! anno divisibile per 4 Calendario Gregoriano ! anno divisibile per 4, se anno del secolo divisibile per 400
Numero solare e Lettera Domenicale Numero solare = (AA + 8) mod 28 + 1
Numero Solare
Lettera Domenicale
Numero Solare
Lettera Domenicale
1 GF 15 C 2 E 16 B 3 D 17 AG 4 C 18 F 5 BA 19 E 6 G 20 D 7 F 21 CB 8 E 22 A 9 DC 23 G 10 B 24 F 11 A 25 ED 12 G 26 C 13 FE 27 B 14 D 28 A
Per gli anni bisestili ci sono due lettere: la prima vale per Gennaio e Febbraio, la seconda per gli altri mesi.
11
Epatta e Numero Aureo Epatta = età della Luna al 1 gennaio - 1. Numero aureo = posizione dell'anno nel ciclo Metonico:
Numero aureo = (AA + 1) mod 19
Num. aureo Epatta 1 29 2 0 3 21 4 2 5 13 6 24 7 5 8 16 9 27
10 8 11 19 12 30 - 0 13 11 14 22 15 3 16 14 17 25 18 6 19 17
Numero del giorno nell'anno dAA = GG del mese + dMM
dMM dMM
Anno Normale
Anno Bisestile
Gennaio 0 0Febbraio 31 31
Marzo 59 60Aprile 90 91
Maggio 120 121Giugno 151 152Luglio 181 182Agosto 212 213
Settembre 243 244Ottobre 273 274
Novembre 304 305Dicembre 334 335
12
Giorno dell'anno Giorno Gen Feb Mar Apr Mag Giu Lug Ago Set Ott Nov Dic
1 1 32 60 91 121 152 182 213 244 274 305 335 2 2 33 61 92 122 153 183 214 245 275 306 336 3 3 34 62 93 123 154 184 215 246 276 307 337 4 4 35 63 94 124 155 185 216 247 277 308 338 5 5 36 64 95 125 156 186 217 248 278 309 339 6 6 37 65 96 126 157 187 218 249 279 310 340 7 7 38 66 97 127 158 188 219 250 280 311 341 8 8 39 67 98 128 159 189 220 251 281 312 342 9 9 40 68 99 129 160 190 221 252 282 313 343 10 10 41 69 100 130 161 191 222 253 283 314 344 11 11 42 70 101 131 162 192 223 254 284 315 345 12 12 43 71 102 132 163 193 224 255 285 316 346 13 13 44 72 103 133 164 194 225 256 286 317 347 14 14 45 73 104 134 165 195 226 257 287 318 348 15 15 46 74 105 135 166 196 227 258 288 319 349 16 16 47 75 106 136 167 197 228 259 289 320 350 17 17 48 76 107 137 168 198 229 260 290 321 351 18 18 49 77 108 138 169 199 230 261 291 322 352 19 19 50 78 109 139 170 200 231 262 292 323 353 20 20 51 79 110 140 171 201 232 263 293 324 354 21 21 52 80 111 141 172 202 233 264 294 325 355 22 22 53 81 112 142 173 203 234 265 295 326 356 23 23 54 82 113 143 174 204 235 266 296 327 357 24 24 55 83 114 144 175 205 236 267 297 328 358 25 25 56 84 115 145 176 206 237 268 298 329 359 26 26 57 85 116 146 177 207 238 269 299 330 360 27 27 58 86 117 147 178 208 239 270 300 331 361 28 28 59 87 118 148 179 209 240 271 301 332 362 29 29 (60 )* 88 119 149 180 210 241 272 302 333 363 30 30 ... 89 120 150 181 211 242 273 303 334 364 31 31 ... 90 ... 151 ... 212 243 ... 304 ... 365
se l'anno è bisestile, dopo il 29 febbraio, aggiungere 1 al giorno dell'anno
13
Data di Pasqua Calcolare:
G = AA mod 19 (X mod Y = resto di X/Y; X\Y = divisione intera)
Per il calendario Giuliano:
I = (19·G + 15) mod 30 J = (AA + AA\4 + I) mod 7
Per il calendario Gregoriano:
C = AA \ 100 H = (C - C\4 - (8·C+13)\25 + 19·G + 15) mod 30 I = H - (H\28)·(1 - (H\28)·(29\(H + 1)) · ((21 - G)\11)) J = (AA + AA\4 + I + 2 - C + C\4) mod 7
Per entrambi i calendari:
L = I - J Mese di Pasqua = 3 + (L + 40) \ 44 Giorno di Pasqua = L + 28 - 31 · (Mese di Pasqua \ 4)
Nota:
G = numero d'oro - 1 H = (23 - epatta) mod 30 I = numero di giorni fra il 21 marzo e il plenilunio pasquale J = giorno della settimana del plenilunio pasquale (dove 0=domenica, 1=lunedì, ecc.) L = numero di giorni dal 21 marzo alla domenica del plenilunio pasquale o precedente il plenilunio (varia fra -6 e 28).
Feste mobili:
Sessagesima: 63 giorni prima Pasqua Quinquagesima: 49 giorni prima Pasqua Ceneri: 46 giorni prima Pasqua Domenica delle Palme: 7 giorni prima Pasqua Venerdì Santo: 2 giorni prima Pasqua Rogazione: 35 giorni dopo Pasqua Ascensione: 39 giorni dopo Pasqua Pentecoste: 49 giorni dopo Pasqua Trinità: 56 giorni dopo Pasqua Corpus Domini: 60 giorni dopo Pasqua
Giorno Giuliano (JD) Numero del giorno (e frazione = ore UT) contato a partire dal 1 gennaio 4713 a.C. ore 12 UT (cioè 1 gennaio -4712 12h UT)
14
Calcolo di JD (X\Y = divisione intera)
JD = ( 1461 · ( AA + 4800 + ( MM - 14 ) \ 12 ) ) \ 4 + ( 367 · ( MM - 2 - 12 · ( ( MM - 14 ) \ 12 ) ) ) \ 12 - ( 3 · ( ( AA + 4900 + ( MM - 14 ) \ 12 ) \ 100 ) ) \ 4 + DD - 32075
Per calcolare la data di calendario a partire da JD:
L = JD + 68569 N = ( 4 · L ) \ 146097 L = L - ( 146097 · N + 3 ) \ 4 I = ( 4000 · ( L + 1 ) ) \ 1461001 L = L - ( 1461 · I ) \ 4 + 31 J = ( 80 · L ) \ 2447 DD = L - ( 2447 · J ) \ 80 L = J \ 11 MM = J + 2 - ( 12 · L ) AA = 100 · ( N - 49 ) + I + L
DD sono interi fra 1-31, MM sono interi fra 1-12, YY sono interi positivi o negativi. X \ Y = divisione intera = parte intera di X/Y (ignorare i decimali)
Giorno Giuliano Modificato MJD = JD - 2 400 000.5
con JD = 0 al 17 novembre 1858 ore 0 UT.
Epoca standard J2000.0 1 gennaio 2000 ore 12 UT ! JD2000 = 2451545.0
Numero di giorni dall'epoca J2000.0 d2000 = JD-JD2000 = JD - 2451544.5
JD approssimato (± 1 giorno):
JD(primo anno del secolo) + int(AA·365.25) + dAA
Giorno della settimana resto di JD / 7 (JD/7 - int(JD/7) · 7
(0=lun, 1=mar, 2=mer, 3=gio, 4=ven, 5=sab, 6=dom)
15
Tabelle per il calcolo di JD SY = anno astronomico + 4712 se MM = Gen o Feb: ! SY = SY - 1 Leggere le componenti per SY e Mese e sommare, aggiungere il giorno del mese. Se la data è Gregoriana, aggiungere la correzione: G
SY JD SY JD Mese JD 8000 2922 059 96 35064 Mar 0 7000 2556 809 84 30681 Apr 31 6000 2191 559 72 26298 Mag 61 5000 1826 309 60 21915 Giu 92 4000 1461 059 48 17532 Lug 122 3000 1095 809 36 13149 Ago 153 2000 7305 59 24 8766 Set 184
800 2922 00 12 4383 Ott 214 600 2191 50 8 2922 Nov 245 400 1461 00 4 1461 Dic 275 300 1095 75 3 1095 Gen 306 200 7305 0 2 730 Feb 337 100 3652 5 1 365
SY G SY G SY G 12 37 2512 18 5012 -1
112 36 2612 17 5212 -2 212 35 2812 16 5312 -3 412 34 2912 15 5412 -4 512 33 3012 14 5612 -5 612 32 3212 13 5712 -6 812 31 3312 12 5812 -7 912 30 3412 11 6012 -8
1012 29 3612 10 6112 -9 1212 28 3712 9 6212 -10 1312 27 3812 8 6412 -11 1412 26 4012 7 6512 -12 1612 25 4112 6 6612 -13 1712 24 4212 5 6812 -14 1812 23 4412 4 6912 -15 2012 22 4512 3 7012 -16 2112 21 4612 2 7212 -17 2212 20 4812 1 7312 -18 2412 19 4912 0 7412 -19
16
Anno JD (1 Gen 0h UT) Anno AA JD
5000 3547 272.5 1 a.C. 0 1721057.54900 3510 748.5 100 a.C. -99 1684898.54800 3474 223.5 200 a.C. -199 1648373.54700 3437 699.5 300 a.C. -299 1611848.54600 3401 175.5 400 a.C. -399 1575323.54500 3364 651.5 500 a.C. -499 1538798.54400 3328 126.5 600 a.C. -599 1502273.54300 3291 602.5 700 a.C. -699 1465748.54200 3255 078.5 800 a.C. -799 1429223.54100 3218 554.5 900 a.C. -899 1392698.54000 3182 029.5 1000 a.C. -999 1356173.53900 3145 505.5 1100 a.C. -1099 1319648.53800 3108 981.5 1200 a.C. -1199 1283123.53700 3072 457.5 1300 a.C. -1299 1246598.53600 3035 932.5 1400 a.C. -1399 1210073.53500 2999 408.5 1500 a.C. -1499 1173548.53400 2962 884.5 1600 a.C. -1599 1137023.53300 2926 360.5 1700 a.C. -1699 1100498.53200 2889 835.5 1800 a.C. -1799 1063973.53100 2853 311.5 1900 a.C. -1899 1027448.53000 2816 787.5 2000 a.C. -1999 990923.52900 2780 263.5 2100 a.C. -2099 954398.52800 2743 738.5 2200 a.C. -2199 917873.52700 2707 214.5 2300 a.C. -2299 881348.52600 2670 690.5 2400 a.C. -2399 844823.52500 2634 166.5 2500 a.C. -2499 808298.52400 2597 641.5 2600 a.C. -2599 771773.52300 2561 117.5 2700 a.C. -2699 735248.52200 2524 593.5 2800 a.C. -2799 698723.52100 2488 069.5 2900 a.C. -2899 662198.52000 2451 544.5 3000 a.C. -2999 625673.51900 2415 020.5 3100 a.C. -3099 589148.51800 2378 496.5 3200 a.C. -3199 552623.51700 2341972.5 15 Ott 1582 3300 a.C. -3299 516098.51600 2305 447.5 2299160.5 3400 a.C. -3399 479573.51500 2268932.5 4 Ott 1582 3500 a.C. -3499 443048.51400 2232 407.5 2299159.5 3600 a.C. -3599 406523.51300 2195 882.5 3700 a.C. -3699 369998.51200 2159357.5 4 Ott 1582 3800 a.C. -3799 333473.51100 2122 832.5 seguito da: 3900 a.C. -3899 296948.51000 2086307.5 15 Ott 1582 4000 a.C. -3999 260423.5
900 2049 782.5 4100 a.C. -4099 223898.5800 2013 257.5 4200 a.C. -4199 187373.5700 1976 732.5 4300 a.C. -4299 150848.5600 1940 207.5 4400 a.C. -4399 114323.5500 1903 682.5 4500 a.C. -4499 77798.5400 1867 157.5 4600 a.C. -4599 41273.5300 1830 632.5 4700 a.C. -4699 4748.5200 1794 107.5 4713 a.C. -4712 0.0100 1757 582.5
1 1721 423.5
17
Astrodate (±0.003, ± 1 giorno):
Data Giuliana: AA + (GG / 365) Data Gregoriana: 0.9999794·(AA +GG / 365) + 0.005
Astrodate = (JD - 1721057.5) / 365.25
Cicli periodici Ciclo solare giuliano: 28 anni; le date del calendario si ripetono negli stessi giorni. Ciclo solare gregoriano: 400 anni; le date del calendario si ripetono negli stessi giorni. Ciclo Metonico: 19 anni; fase della Luna si ripete nelle stesse date di calendario. Saros: 6585.321 (6585 giorni e 1/3) (8 anni, 11 giorni e 1/3 o 10 e 1/3 se nel periodo ci sono 5 anni bisestili, 12 e 1/3 se ce ne sono tre); si ripete la stessa configurazione Sole/Luna e si ha un'eclissi di Sole simile alla precedente. Ciclo di Saros: circa 80 eclissi. Viene da: 223 mesi sinodici (6585.321 giorni) ~ 242 mesi draconici (6585.357 giorni) ~ 19 anni dell'eclissi (6585.781 giorni).
18
Altri calendari
Calendario Data d'inizio
Num. giorni per anno
Num. mesi per anno
Correzioni
Astronomico ≈365.24219 Azteco Sacro 260 20 di 13 giorni Azteco Solare 365 18 di 20 giorni
+ 5 giorni
Bahai 21 mar 1844 d.C.
365.2425 19 di 19 giorni e 4 o 5 giorni festivi
se l'anno è divisibile per 4 è bisestile se l'anno è divisibile per 100 non è bisestile se l'anno è divisibile per 400 è bisestile
Cinese 353 ÷ 385 12 o 13 di 29 o 30 giorni
un mese intercalare aggiunto con regole complesse
Copto 365.2425 12 di 30 giorni + 5 o 6 giorni
se l'anno è divisibile per 4 è bisestile se l'anno è divisibile per 100 non è bisestile se l'anno è divisibile per 400 è bisestile
Egiziano 365 12 di 30 giorni + 5 giorni
Inglese 12 Francese Rivoluzionario
22 set 1792 d.C.
365.24225 12 di 30 giorni + 5 o 6 giorni
anni 3, 7 e 11 sono bisestili Calendario abolito nell'anno 14.
Greco Ortodosso
365.24222 12 in totale 7 x 31, 4 x 30, 1 x 28 o 29
se l'anno è divisibile per 4 è bisestile se l'anno è divisibile per 100 non è bisestile se l'anno diviso per 900 dà come resto 200 o 600 è bisestile
Gregoriano in teoria: 1 gen 1 d.C. (vedi tab.)
365.2425 12 in totale 7 x 31, 4 x 30, 1 x 28 o 29
se l'anno è divisibile per 4 è bisestile se l'anno è divisibile per 100 non è bisestile se l'anno è divisibile per 400 è bisestile
Ebreo 3761 a.C. ≈354 12 o 13 di 29 o 30 giorni.
se l'anno diviso per 19 dà come resto 0, 3, 6, 8, 11, 14 o 17 è bisestile ed ha 13 mesi.
Ebreo del giubileo
364 12 di 30 giorni + 1 extra in ogni trimestre
Indiano 22 mar 79 d.C.
365.2425 12 in totale 5 x 31 7 x 30
se l'anno è divisibile per 4 è bisestile se l'anno è divisibile per 100 non è bisestile se l'anno è divisibile per 400 è bisestile
Islamico (Egira)
16 lug 622 d.C.
354.36 12 di lunghezza variabile
basato sull'osservazione della prima luna ogni mese
Giuliano 45 a.C. 365.25 12 in totale 7 x 31, 4 x 30, 1 x 28 o 29
se l'anno è divisibile per 4 è bisestile
Lunare 354 12 in totale 6 x 29, 6 x 30
Maya Sacro 12 ago 3113 a.C.
260 20 di 13 giorni
Maya Solare 12 ago 3113 a.C.
365 18 di 20 giorni + 5 extra giorni
Persiano 21 mar 622 d.C.
365.2422 12 in totale 5 x 30, 6 x 31 1 x 29 o 30
un giorno extra inserito secondo un complesso ciclo che dura 2820 anni
Romano 750 a.C. Standard: 355 Media: 366.25
12 in totale 1 x 28, 7 x 29, 4 x 31
un mese addizionale di 28 o 29 giorni in anni alternati
19
Coordinate celesti
Trasformazione di coordinate
Orizzontali ed Equatoriali A = azimuth, h = altezza HA = angolo orario δ = declinazione, α = ascensione retta LST = Tempo siderale locale ϕ = Latitudine geografica
Calcolare: HA = LST - α
orizzontali ! equatoriali:
cos δ · sin HA = cos h · sin A sin δ = sin ϕ · sin h - cos ϕ · cos h · cos A
cos δ · cos HA = cos ϕ · sin h + sin ϕ · cos h · cos A equatoriali ! orizzontali
cos h · sin A = cos δ · sin HA sin h = sin ϕ · sin δ + cos ϕ · cos δ · cos HA
cos h · cos A = - cos ϕ · sin δ + sin ϕ · cos δ · cos HA
Equatoriali e Eclittiche β = latitudine celeste, λ = longitudine celeste ε = obliquità dell'eclittica.
equatoriali ! eclittiche:
cos β · cos λ = cos δ · cos α cos β · sin λ = cos δ · sin α · cos ε + sin δ · sin ε
sin β = - cos δ · sin α · sin ε + sin δ · cos ε eclittiche ! equatoriali:
cos δ · cos α = cos β · cos λ cos δ · sin α = cos β · sin λ · cos ε - sin β · sin ε
sin δ = cos β · sin λ · sin ε + sin β · cos ε
Altezza e Azimuth
Calcolare: HA = LST - α sin h = sin ϕ · sin δ + cos ϕ · cos δ · cos HA
20
ϕ⋅−ϕ⋅=
cosδtansinHAcossinHAAtan
dove:
h = altezza A = azimuth (A = 0° = SUD, A = 90° = OVEST) HA = angolo orario L'arcotangente va calcolata nel quadrante giusto, vedi sezione Trigonometria
Altezza sull'orizzonte al meridiano (ore 12 locali) in gradi di un oggetto con declinazione δ, in luogo con latitudine ϕ:
h = 90° - ϕ + δ
Azimuth dei punti di sorgere e tramonto cos A0 = - sin δ/ cos ϕ
Declinazione di un oggetto sull'eclittica tan δ = sin α · tan ε
Ora approssimata di alba / tramonto ora UT = 12 - Et + (L + k·arccos(tan δ · tan ϕ)) / 15
(k=-1 alba, k=1 tramonto, L = longitudine).
Correzione all'altezza per rifrazione R0 = tan (90 - 0.99914·HC - 7.31 / (H + 4.4))
R0 è in minuti, HC = altezza calcolata in gradi, R0 vale a 1010 mbar e 10°C, altrimenti:
f = 0.28·P / (T + 273°)
dove P = pressione atmosferica in mbar, T = temperatura in °C quindi
R = R0·f L'altezza vera sarà: H = HC + R La regola di Thom-Neugebauer dice che una stella di magnitudine visuale "m" diverra' visibile nel cielo mattutino ad un'altezza sull'orizzonte approssimativamente pari alla sua magnitudine visuale.
Allineamenti Verifica di possibili allineamenti archeoastronomici.
21
r = 0.58 · e(-H/8400) · e(-S/3) G = S r + p sin d = sin L · sin G + cos L · cos G · cos A sin B = cos L · sin A / cos δ ∆δ = δ + q · cos B d V = ∆δ · sec B · (1-r/3)
dove:
S = altezza (°) dell'orizzonte sul piano δ = declinazione obiettivo (stella, Sole, Luna) A = azimuth misurato da N verso E L = latitudine del luogo H = altezza s.l.m. in metri q = semidiametro del disco dell'obiettivo p = parallasse dell'obiettivo
Semidiametro: 0 stella o centro disco -0.267 sole tang sopra (emisf. Nord) 0.267 sole tang sotto (emisf. Nord) -0.259 luna tang sopra (emisf. Nord) 0.259 luna tang sotto (emisf. Nord)
Parallasse: 0 stella 0.002 sole 0.951 luna
r = angolo di rifrazione V = angolo di quanto l'obiettivo è sopra la linea dell'orizzonte in direzione dell'azimuth della
linea di osservazione ∆δ = errore in declinazione
Potere risolutivo dell'occhio Potere risolutivo dell'occhio nudo in funzione della magnitudine visuale M1 e M2 di due stelle osservate:
)ee(28".240)("p 2M46.01M46.0 −− +⋅= Poiché i recettori sulla retina sono spaziati di 30", questo valore è un il limite inferiore per il potere risolutivo dell'occhio nudo. I valori di p" minori di 30" d'arco non vanno presi in considerazione. Potere risolutivo dell'occhio nudo in funzione della magnitudine visuale delle due stelle.
M1 M2 p" p' p° -3 -3 1910.18 31.84 0.53 -3 -2 1558.02 25.97 0.43 -3 -1 1335.71 22.26 0.37 -3 0 1195.37 19.92 0.33 -3 1 1106.77 18.45 0.31 -3 2 1050.85 17.51 0.29 -3 3 1015.54 16.93 0.28
22
M1 M2 p" p' p° -3 4 993.25 16.55 0.28 -2 -2 1205.86 20.10 0.33 -2 -1 983.55 16.39 0.27 -2 0 843.21 14.05 0.23 -2 1 754.62 12.58 0.21 -2 2 698.69 11.64 0.19 -2 3 663.38 11.06 0.18 -2 4 641.09 10.68 0.18 -1 -1 761.24 12.69 0.21 -1 0 620.90 10.35 0.17 -1 1 532.31 8.87 0.15 -1 2 476.38 7.94 0.13 -1 3 441.07 7.35 0.12 -1 4 418.78 6.98 0.12 0 0 480.56 8.01 0.13 0 1 391.96 6.53 0.11 0 2 336.04 5.60 0.09 0 3 300.73 5.01 0.08 0 4 278.44 4.64 0.08 1 1 303.37 5.06 0.08 1 2 247.44 4.12 0.07 1 3 212.13 3.54 0.06 1 4 189.85 3.16 0.05 2 2 191.51 3.19 0.05 2 3 156.21 2.60 0.04 2 4 133.92 2.23 0.04 3 3 120.90 2.02 0.03 3 4 98.61 1.64 0.03 4 4 76.32 1.27 0.02
Semiarco Diurno semiarco (gradi) = arcos(-tan δ · tan ϕ)
23
Semiarco diurno (ore)
valori in hh.mm
Declin Lat.Geo 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0
90 6.0085 6.0080 12.00 7.59 6.0075 12.00 8.45 7.16 6.0070 12.00 9.10 7.56 6.56 6.0065 12.00 9.25 8.20 7.29 6.43 6.0060 12.00 9.35 8.36 7.51 7.11 6.35 6.0055 12.00 9.42 8.47 8.05 7.30 6.58 6.29 6.0050 12.00 9.46 8.54 8.15 7.43 7.14 6.49 6.24 6.0045 12.00 9.48 8.58 8.21 7.51 7.25 7.02 6.41 6.20 6.0040 12.00 9.48 8.59 8.24 7.56 7.32 7.11 6.52 6.34 6.17 6.0035 12.00 9.46 8.58 8.24 7.57 7.35 7.16 6.59 6.43 6.28 6.14 6.0030 12.00 9.42 8.54 8.21 7.56 7.35 7.18 7.02 6.49 6.36 6.23 6.12 6.0025 12.00 9.35 8.47 8.15 7.51 7.32 7.16 7.02 6.50 6.39 6.29 6.19 6.09 6.0020 12.00 9.25 8.36 8.05 7.43 7.25 7.11 6.59 6.49 6.39 6.30 6.22 6.15 6.07 6.0015 12.00 9.10 8.20 7.51 7.30 7.14 7.02 6.52 6.43 6.36 6.29 6.22 6.16 6.11 6.05 6.0010 12.00 8.45 7.56 7.29 7.11 6.58 6.49 6.41 6.34 6.28 6.23 6.19 6.15 6.11 6.07 6.04 6.00
5 7.59 7.16 6.56 6.43 6.35 6.29 6.24 6.20 6.17 6.14 6.12 6.09 6.07 6.05 6.04 6.02 6.000 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00
-5 4.01 4.44 5.04 5.17 5.25 5.31 5.36 5.40 5.43 5.46 5.48 5.51 5.53 5.55 5.56 5.58 6.00-10 0.00 3.15 4.04 4.31 4.49 5.02 5.11 5.19 5.26 5.32 5.37 5.41 5.45 5.49 5.53 5.56 6.00-15 0.00 2.50 3.40 4.09 4.30 4.46 4.58 5.08 5.17 5.24 5.31 5.38 5.44 5.49 5.55 6.00-20 0.00 2.35 3.24 3.55 4.17 4.35 4.49 5.01 5.11 5.21 5.30 5.38 5.45 5.53 6.00-25 0.00 2.25 3.13 3.45 4.09 4.28 4.44 4.58 5.10 5.21 5.31 5.41 5.51 6.00-30 0.00 2.18 3.06 3.39 4.04 4.25 4.42 4.58 5.11 5.24 5.37 5.48 6.00-35 0.00 2.14 3.02 3.36 4.03 4.25 4.44 5.01 5.17 5.32 5.46 6.00-40 0.00 2.12 3.01 3.36 4.04 4.28 4.49 5.08 5.26 5.43 6.00-45 0.00 2.12 3.02 3.39 4.09 4.35 4.58 5.19 5.40 6.00-50 0.00 2.14 3.06 3.45 4.17 4.46 5.11 5.36 6.00-55 0.00 2.18 3.13 3.55 4.30 5.02 5.31 6.00-60 0.00 2.25 3.24 4.09 4.49 5.25 6.00-65 0.00 2.35 3.40 4.31 5.17 6.00-70 0.00 2.50 4.04 5.04 6.00-75 0.00 3.15 4.44 6.00-80 0.00 4.01 6.00-85 6.00-90 6.00
Declin Lat.Geo 0 -5 -10 -15 -20 -25 -30 -35 -40 -45 -50 -55 -60 -65 -70 -75 -80 -85
90 6.00 85 6.00 80 6.00 4.01 0.00 75 6.00 4.44 3.15 0.00 70 6.00 5.04 4.04 2.50 0.00 65 6.00 5.17 4.31 3.40 2.35 0.00 60 6.00 5.25 4.49 4.09 3.24 2.25 0.00 55 6.00 5.31 5.02 4.30 3.55 3. 13 2.18 0.00 50 6.00 5.36 5.11 4.46 4.17 3.45 3.06 2.14 0.00 45 6.00 5.40 5.19 4.58 4.35 4.09 3.39 3.02 2.12 0.00 40 6.00 5.43 5.26 5.08 4.49 4.28 4.04 3.36 3.01 2.12 0.00 35 6.00 5.46 5.32 5.17 5.01 4. 44 4.25 4.03 3.36 3.02 2.14 0.00 30 6.00 5.48 5.37 5.24 5.11 4.58 4.42 4.25 4.04 3.39 3.06 2.18 0.00 25 6.00 5.51 5.41 5.31 5.21 5.10 4.58 4.44 4.28 4.09 3.45 3.13 2.25 0.00 20 6.00 5.53 5.45 5.38 5.30 5.21 5.11 5.01 4.49 4.35 4.17 3.55 3.24 2.35 0.00 15 6.00 5.55 5.49 5.44 5.38 5.31 5.24 5.17 5.08 4.58 4.46 4.30 4.09 3.40 2.50 0.00 10 6.00 5.56 5.53 5.49 5.45 5.41 5.37 5.32 5.26 5.19 5.11 5.02 4.49 4.31 4.04 3.15 0.00 5 6.00 5.58 5.56 5.55 5.53 5.51 5.48 5.46 5.43 5.40 5.36 5.31 5.25 5.17 5.04 4.44 4.01 0 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00
-5 6.00 6.02 6.04 6.05 6.07 6.09 6.12 6.14 6.17 6.20 6.24 6.29 6.35 6.43 6.56 7.16 7.59 -10 6.00 6.04 6.07 6.11 6.15 6.19 6.23 6.28 6.34 6.41 6.49 6.58 7.11 7.29 7.56 8.45 12.00 -15 6.00 6.05 6.11 6.16 6.22 6.29 6.36 6.43 6.52 7.02 7.14 7.30 7.51 8.20 9.10 12.00 -20 6.00 6.07 6.15 6.22 6.30 6.39 6.49 6.59 7.11 7.25 7.43 8.05 8.36 9.25 12.00 -25 6.00 6.09 6.19 6.29 6.39 6.50 7.02 7.16 7.32 7.51 8.15 8.47 9.35 12.00 -30 6.00 6.12 6.23 6.36 6.49 7.02 7.18 7.35 7.56 8.21 8.54 9.42 12.00 -35 6.00 6.14 6.28 6.43 6.59 7. 16 7.35 7.57 8.24 8.58 9.46 12.00 -40 6.00 6.17 6.34 6.52 7.11 7.32 7.56 8.24 8.59 9.48 12.00 -45 6.00 6.20 6.41 7.02 7.25 7.51 8.21 8.58 9.48 12.00 -50 6.00 6.24 6.49 7.14 7.43 8.15 8.54 9.46 12.00 -55 6.00 6.29 6.58 7.30 8.05 8. 47 9.42 12.00 -60 6.00 6.35 7.11 7.51 8.36 9.35 12.00 -65 6.00 6.43 7.29 8.20 9.25 12.00 -70 6.00 6.56 7.56 9.10 12.00 -75 6.00 7.16 8.45 12.00 -80 6.00 7.59 12.00 -85 6.00 -90 6.00
MAI VISIBILE
SEMPRE VISIBILE
MAI VISIBILE
SEMPRE VISIBILE
24
Precessione
Precessione approssimata Per coordinate equatoriali, la precessione annuale è:
p_α = m + n · sin α · tan δ p_δ = n · cos α
Quindi per ridurre le posizioni dall'anno Y all'anno X :
αX = αY + p_α · t δX = δY + p_δ · t
dove t è il tempo trascorso fra X e Y in anni.
(per l'anno 2000, m e n sono: m = + 46.124" + 0.000279" · t = 3.0749 s + 0.0000186 s · t n = + 20.043" - 0.000085" · t
t tempo in anni tropici dal 2000.0) Per coordinate eclittiche i valori approssimati sono dati da:
λX = λY + PrecLon · t δX ≅ δY
dove t è il tempo trascorso fra X e Y in anni.
(per l'anno 2000: PrecLon = 50.28792"). La validità di queste fomule è di qualche decina di anni. Le tabelle che seguono contengono i valori di m, n e della precessione generale in longitudine calcolati per ogni inizio secolo dal 4713 a.C. al 5000 d.C.
25
Coefficienti per il calcolo della precessione Anno (1 Gen 0h UT) JD m
arcsec m sec
n arcsec
n sec
PrecLon ("/anno)
5000 d.C. 3547272.5 46.93498 3.12900 19.79409 1.3196 1 50.619054900 d.C. 3510748.5 46.90961 3.12731 19.80188 1.3201 3 50.608024800 d.C. 3474223.5 46.88408 3.12561 19.80974 1.3206 5 50.596984700 d.C. 3437699.5 46.85838 3.12389 19.81764 1.3211 8 50.585944600 d.C. 3401175.5 46.83254 3.12217 19.82559 1.3217 1 50.574904500 d.C. 3364651.5 46.80654 3.12044 19.83359 1.3222 4 50.563874400 d.C. 3328126.5 46.78040 3.11869 19.84164 1.3227 8 50.552834300 d.C. 3291602.5 46.75412 3.11694 19.84972 1.3233 1 50.541794200 d.C. 3255078.5 46.72771 3.11518 19.85785 1.3238 6 50.530754100 d.C. 3218554.5 46.70117 3.11341 19.86602 1.3244 0 50.519714000 d.C. 3182029.5 46.67451 3.11163 19.87423 1.3249 5 50.508683900 d.C. 3145505.5 46.64774 3.10985 19.88247 1.3255 0 50.497643800 d.C. 3108981.5 46.62085 3.10806 19.89074 1.3260 5 50.486603700 d.C. 3072457.5 46.59386 3.10626 19.89905 1.3266 0 50.475563600 d.C. 3035932.5 46.56678 3.10445 19.90738 1.3271 6 50.464523500 d.C. 2999408.5 46.53959 3.10264 19.91574 1.3277 2 50.453493400 d.C. 2962884.5 46.51232 3.10082 19.92413 1.3282 8 50.442453300 d.C. 2926360.5 46.48497 3.09900 19.93254 1.3288 4 50.431413200 d.C. 2889835.5 46.45754 3.09717 19.94097 1.3294 0 50.420373100 d.C. 2853311.5 46.43004 3.09534 19.94943 1.3299 6 50.409343000 d.C. 2816787.5 46.40247 3.09350 19.95789 1.3305 3 50.398302900 d.C. 2780263.5 46.37484 3.09166 19.96638 1.3310 9 50.387262800 d.C. 2743738.5 46.34716 3.08981 19.97488 1.3316 6 50.376222700 d.C. 2707214.5 46.31942 3.08796 19.98339 1.3322 3 50.365182600 d.C. 2670690.5 46.29165 3.08611 19.99191 1.3327 9 50.354152500 d.C. 2634166.5 46.26383 3.08426 20.00043 1.3333 6 50.343112400 d.C. 2597641.5 46.23598 3.08240 20.00896 1.3339 3 50.332072300 d.C. 2561117.5 46.20810 3.08054 20.01750 1.3345 0 50.321032200 d.C. 2524593.5 46.18020 3.07868 20.02604 1.3350 7 50.310002100 d.C. 2488069.5 46.15229 3.07682 20.03457 1.3356 4 50.298962000 d.C. 2451544.5 46.12436 3.07496 20.04311 1.3362 1 50.287921900 d.C. 2415020.5 46.09643 3.07310 20.05164 1.3367 8 50.276881800 d.C. 2378496.5 46.06850 3.07123 20.06016 1.3373 4 50.265841700 d.C. 2341972.5 46.04057 3.06937 20.06868 1.3379 1 50.254811600 d.C. 2305447.5 46.01265 3.06751 20.07719 1.3384 8 50.243771500 d.C. 2268932.5 45.98476 3.06565 20.08568 1.3390 5 50.232731582 d.C. Oct 15 2299160.5 46.00785 3.06719 20.07865 1.3385 8 50.241871582 d.C. Oct 04 2299159.5 46.00785 3.06719 20.07865 1.3385 8 50.241871400 d.C. 2232407.5 45.95688 3.06379 20.09416 1.3396 1 50.221701300 d.C. 2195882.5 45.92903 3.06194 20.10262 1.3401 7 50.210661200 d.C. 2159357.5 45.90121 3.06008 20.11106 1.3407 4 50.199621100 d.C. 2122832.5 45.87343 3.05823 20.11949 1.3413 0 50.188581000 d.C. 2086307.5 45.84570 3.05638 20.12789 1.3418 6 50.17754
900 d.C. 2049782.5 45.81801 3.05453 20.13627 1.3424 2 50.16650800 d.C. 2013257.5 45.79038 3.05269 20.14462 1.3429 7 50.15547700 d.C. 1976732.5 45.76281 3.05085 20.15294 1.3435 3 50.14443600 d.C. 1940207.5 45.73530 3.04902 20.16124 1.3440 8 50.13339500 d.C. 1903682.5 45.70787 3.04719 20.16950 1.3446 3 50.12235400 d.C. 1867157.5 45.68051 3.04537 20.17773 1.3451 8 50.11131300 d.C. 1830632.5 45.65324 3.04355 20.18592 1.3457 3 50.10028200 d.C. 1794107.5 45.62606 3.04174 20.19407 1.3462 7 50.08924100 d.C. 1757582.5 45.59896 3.03993 20.20219 1.3468 1 50.07820
1 d.C. 1721423.5 45.57224 3.03815 20.21018 1.3473 5 50.06727
26
Anno (1 Gen 0h UT) JD m
arcsec m sec
n arcsec
n sec
PrecLon ("/anno)
1 a.C. 0 1721057.5 45.57197 3.03813 20.21026 1.3473 5 50.06716100 a.C. -99 1684898.5 45.54535 3.03636 20.21821 1.3478 8 50.05623200 a.C. -199 1648373.5 45.51857 3.03457 20.22619 1.3484 1 50.04520300 a.C. -299 1611848.5 45.49191 3.03279 20.23413 1.3489 4 50.03416400 a.C. -399 1575323.5 45.46536 3.03102 20.24201 1.3494 7 50.02312500 a.C. -499 1538798.5 45.43895 3.02926 20.24984 1.3499 9 50.01208600 a.C. -599 1502273.5 45.41266 3.02751 20.25762 1.3505 1 50.00104700 a.C. -699 1465748.5 45.38651 3.02577 20.26535 1.3510 2 49.99001800 a.C. -799 1429223.5 45.36051 3.02403 20.27301 1.3515 3 49.97897900 a.C. -899 1392698.5 45.33465 3.02231 20.28062 1.3520 4 49.96793
1000 a.C. -999 1356173.5 45.30895 3.02060 20.28816 1.3525 4 49.956891100 a.C. -1099 1319648.5 45.28341 3.01889 20.29564 1.3530 4 49.945851200 a.C. -1199 1283123.5 45.25803 3.01720 20.30305 1.3535 4 49.934821300 a.C. -1299 1246598.5 45.23283 3.01552 20.31040 1.3540 3 49.923781400 a.C. -1399 1210073.5 45.20780 3.01385 20.31768 1.3545 1 49.912741500 a.C. -1499 1173548.5 45.18295 3.01220 20.32488 1.3549 9 49.901701600 a.C. -1599 1137023.5 45.15829 3.01055 20.33202 1.3554 7 49.890661700 a.C. -1699 1100498.5 45.13382 3.00892 20.33907 1.3559 4 49.879631800 a.C. -1799 1063973.5 45.10955 3.00730 20.34605 1.3564 0 49.868591900 a.C. -1899 1027448.5 45.08548 3.00570 20.35295 1.3568 6 49.857552000 a.C. -1999 990923.5 45.06162 3.00411 20.35977 1.3573 2 49.846512100 a.C. -2099 954398.5 45.03798 3.00253 20.36650 1.3577 7 49.835472200 a.C. -2199 917873.5 45.01455 3.00097 20.37315 1.3582 1 49.824442300 a.C. -2299 881348.5 44.99135 2.99942 20.37971 1.3586 5 49.813402400 a.C. -2399 844823.5 44.96839 2.99789 20.38618 1.3590 8 49.802362500 a.C. -2499 808298.5 44.94566 2.99638 20.39256 1.3595 0 49.791322600 a.C. -2599 771773.5 44.92317 2.99488 20.39885 1.3599 2 49.780282700 a.C. -2699 735248.5 44.90093 2.99340 20.40504 1.3603 4 49.769252800 a.C. -2799 698723.5 44.87894 2.99193 20.41113 1.3607 4 49.758212900 a.C. -2899 662198.5 44.85721 2.99048 20.41713 1.3611 4 49.747173000 a.C. -2999 625673.5 44.83575 2.98905 20.42302 1.3615 3 49.736133100 a.C. -3099 589148.5 44.81455 2.98764 20.42881 1.3619 2 49.725093200 a.C. -3199 552623.5 44.79363 2.98624 20.43450 1.3623 0 49.714063300 a.C. -3299 516098.5 44.77300 2.98487 20.44007 1.3626 7 49.703023400 a.C. -3399 479573.5 44.75264 2.98351 20.44554 1.3630 4 49.691983500 a.C. -3499 443048.5 44.73258 2.98217 20.45090 1.3633 9 49.680943600 a.C. -3599 406523.5 44.71282 2.98085 20.45614 1.3637 4 49.669903700 a.C. -3699 369998.5 44.69336 2.97956 20.46127 1.3640 8 49.658873800 a.C. -3799 333473.5 44.67420 2.97828 20.46628 1.3644 2 49.647833900 a.C. -3899 296948.5 44.65537 2.97702 20.47117 1.3647 4 49.636794000 a.C. -3999 260423.5 44.63685 2.97579 20.47594 1.3650 6 49.625754100 a.C. -4099 223898.5 44.61865 2.97458 20.48059 1.3653 7 49.614714200 a.C. -4199 187373.5 44.60079 2.97339 20.48511 1.3656 7 49.603684300 a.C. -4299 150848.5 44.58326 2.97222 20.48951 1.3659 7 49.592644400 a.C. -4399 114323.5 44.56607 2.97107 20.49377 1.3662 5 49.581604500 a.C. -4499 77798.5 44.54923 2.96995 20.49791 1.3665 3 49.570564600 a.C. -4599 41273.5 44.53274 2.96885 20.50191 1.3667 9 49.559524700 a.C. -4699 4748.5 44.51661 2.96777 20.50577 1.3670 5 49.548484713 a.C. -4712 0.0 44.51454 2.96764 20.50627 1.3670 8 49.54705
27
Sole, Luna e Pianeti
Posizioni di Sole, Luna e Pianeti Calcolare:
AGE = Lungh · PosFrac((Astrodate - Inizio) / Periodo) dove:
Lungh (DD) Periodo Inizio Mercurio 116 0.3172554 0.333 Venere 584 1.598690 0.442 Marte 780 2.135350 1.721 Giove 399 1.092085 0.703 Saturno 378 1.035160 0.41 Luna 29.5 0.08085034 0.4694 Ciclo Lunare (AA) 18.61 18.614 14.9 Long. Sole (gradi) 360 0.9999786 0.223
Eventi Sinodici AGE Mercurio Venere
Congiunzione superiore 0 0 Stella della sera 20 39 Max elongazione alla sera 36 221 Max luminosita 42 257 Scompare come Stella della sera 48 285 Congiunzione inferiore 58 292 Stella del mattino 68 299 Max luminosita 74 327 Max elongazione al mattino 80 362 Scompare come Stella del mattino 104 545 Congiunzione superiore 116 584
Marte Giove Saturno Congiunzione superiore 0 0 0 Stella del mattino 65 20 20 Inizia moto retrogrado 353 140 125 Opposizione 390 200 189 Termina moto retrogrado 427 260 253 Scompare come Stella della sera 725 379 358 Congiunzione superiore 780 399 378
Luna Nuova (congiunzione) 0 Falce serale 1 Primo Quarto 7 Piena 15 Ultimo Quarto 22 Falce al mattino 28 Nuova (congiunzione) 29.5
Long. Sole
28
Eventi Sinodici AGE Equinozio di primavera 0 Solstizio d'estate 90 Equinozio d'autunno 180 Solstizio d'inverno 270
Massima declinazione della Luna
δ(max luna) = 24 + 5 cos(360·PosF((AA-14.9)/18.614)
(PosF(X) = frac(X) se X>0, 1-frac(X) se X<0)
Età della Luna
frac(JD/29.530588) ~0.33 = luna nuova ~0.83 = luna piena
oppure: Età della Luna (±2 gg) = Epatta + numero del mese + numero del giorno
dove: numero del mese = 1 per gennaio, 2 per febbraio, 1 per marzo fino a 10 per dicembre.
Numero di giorni dopo LunaNuova:
Ngg = frac(JD / 29.530588 - 0.33) · 29.5
Posizione del Sole Siano:
frac(7.21) = 0.21; ma frac(-1.2) = 1- frac|-1.2|
T = 0.01*(AA - 2000 + (dAA - 1)/365.25) T = (JD - 2451545.0)/36525.0
Calcolare:
M = 2π · frac(0.993133+99.97361 · T) DL = 6893.0 · sin M + 72.0 · sin 2M L = 2π · frac(0.7859453 + M/2π + (6191.2 · T + DL) / 1 296 000) X = cos L Y = 0.91748 · sin L Z = 0.39778 · sin L ________ r = √ (1.0 - Z2)
allora: δ(sole) = (360/2π) · arctan(Z/r) α(sole) = (48.0/2π) · arctan(Y/(X+r))
δ in gradi, α in ore (ridurre modulo 24).
29
N.B. Le arcotangenti vanno calcolate nel quadrante corretto, vedi sezione sulla trigonometria.
Declinazione approssimata del Sole
δ = -7.64 · sin(dAA - 2) + 9.86 · sin(2 · (dAA - 80)) dove:
dAA = numero di giorni dell'anno e sin() richiede giorni come argomento.
Oppure, utilizzando AGE: δ(sole) = arcsin(sin AGE · sin δ(max sole))
Massima declinazione del Sole
δ(max sole) = 23.7 - 0.00013 · AA
Equazione del Tempo EdT = α(Sole vero) - α(Sole medio)
EdT = 9.87 · sin 2B - 7.53 · cos B - 1.5 · sin B dove:
B = 360 · (dAA - 81) / 365 e dAA = numero di giorni dell'anno: 1 gennaio = giorno 1
Posizioni approssimate del Sole α δ λ EdT(min)Gen 1 18°44' -23.1 280 -3.30
16 19°49' -21.1 295 -9.60Feb 1 20°56' -17.3 312 -13.50
16 21°56' -12.6 327 -14.30Mar 1 22°46' -7.9 340 -12.60
16 23°41' -2.0 355 -8.90Apr 1 0°40' 4.3 11 -4.20
16 1°34' 9.8 26 0.00Mag 1 2°31' 14.9 40 2.80
16 3°29' 18.9 55 3.70Giu 1 4°34' 21.9 70 2.40
16 5°35' 23.3 84 -0.40Lug 1 6°38' 23.2 99 -3.60
16 7°39' 21.5 113 -5.90Ago 1 8°43' 18.2 128 -6.30
16 9°40' 14.0 143 -4.40Set 1 10°39' 8.5 158 -0.20
16 11°33' 2.9 173 4.80Ott 1 12°27' -2.9 187 10.10
16 13°21' -8.6 202 14.20Nov 1 14°23' -14.2 218 16.30
30
16 15°23' -18.6 233 15.30Dic 1 16°26' -21.7 248 11.20
16 17°32' -23.3 264 4.70
Altezza del Sole durante il giorno
hSole = arccos (cos (90 - ϕ) · cos (90-δ) + sin (90 ϕ) · sin (90 δ) · cos (ora ·15)) ora = ora dal mezzogiorno.
Apogeo (Auge) Solare
Anno JD λ° 1200 2159 358 89.21 89°13'1250 2177 621 90.07 90°04'1300 2195 883 90.92 90°55'1350 2214 146 91.78 91°47'1400 2232 408 92.64 92°38'1450 2250 671 93.49 93°30'1500 2268 933 94.35 94°21'1550 2287 196 95.21 95°13'1600 2305 448 96.07 96°04'1650 2323 711 96.92 96°55'1700 2341 973 97.78 97°47'1750 2360 235 98.64 98°38'1800 2378 497 99.50 99°30'1850 2396 759 100.36 100°22'1900 2415 021 101.22 101°13'1950 2433 283 102.08 102°05'2000 2451 545 102.94 102°56'
Longitudine = 102.937348 + (1.7195269·T) + (0.00045962·T2) T = (JD-2451545)/36525
31
Delta T
(1) prima del 948 AD ∆T (sec) = 1830 - 405·T + 46.5·T2 (T = secoli dal 948 AD) (2) dal 948 AD al 1600 AD (T = secoli dal 1850 AD) ∆Τ (sec) = 22.5·T2
∆T (sec)
Anno (1) (2) (3) (4) -2000 4275 7 5418 1 - --1900 4052 4 5108 1 - --1800 3835 0 4807 3 - --1700 3623 6 4515 9 - --1600 3418 1 4233 8 - --1500 3218 7 3961 0 - --1400 3025 3 3697 5 - --1300 2837 8 3443 3 - --1200 2656 4 3198 4 - --1100 2480 9 2962 7 - --1000 2311 5 2736 4 - --900 2148 0 2519 4 - --800 1990 5 2311 7 - --700 1839 0 2113 3 - --600 1693 5 1924 2 - --500 1553 9 1744 4 - 16800-400 1420 4 1573 8 - 15300-300 1292 9 1412 6 - 14000-200 1171 3 1260 7 - 12800-100 1055 7 1118 1 - 11600
0 9462 9848 - 10600100 8426 8608 - 9600200 7450 7461 - 8600300 6534 6406 - 7700400 5678 5445 - 6700500 4882 4577 - 5700600 4145 3802 - 4700700 3469 3120 - 3800800 2852 2531 - 3000900 2296 2035 - 2200
1000 1799 - 1625 16001100 1362 - 1265 11001200 985 - 950 7501300 668 - 680 4701400 411 - 455 3001500 214 - 275 1801600 76 - 140 1101700 -1 - - - 9 osservato 1800 -19 - - - 13.7 osservato 1900 24 - - - -2.7 osservato 2000 126 - - - 64 osservato
32
Fino al 1985, ∆T era calcolato : (T = secoli dal 1900.0): ∆T(sec) = 24.349 + 72.318·T + 29.950·T2 +
Riferimenti: (1): Formula derivata da osservazioni dal 1650 AD di Spencer Jones (1939) e Gerald M.Clemence (1948), accettata ufficialmente dall'International Astronomical Union nel 1952 e leggermente corretta nel 1960. (2) e (3): Stephenson F.R e Houlden M.A., Atlas of Historical Eclipse Maps: East Asia 1500 BD - AD 1900, Cambridge Univ.Press., 1986. (4): Stephenson F.R., Historical Eclipses and Earth's Rotation , Cambridge Univ.Press, 1997. Morrison, L.V. e Ward, C. G., An analysis of the transits of Mercury: 1677-1973, Mon. Not. Roy. Astron. Soc., 173, 183-206, 1975.
33
Coordinate delle stelle più brillanti (J2000.0)
Num Bay Cost Nome ** Vmag RAdec Decdec Lon Lat ARh ARm ARs DE- DEd DEm DEs1 Alp CMa Sirius x -1.46 6.75 -16.72 104.08 -39.61 6 45 8.9 - 16 42 582 Alp Boo Arcturus x -0.04 14.26 19.18 204.23 30.74 14 15 39.7 + 19 10 573 Alp Lyr Vega x 0.03 18.62 38.78 285.32 61.73 18 36 56.3 + 38 47 14 Alp Aur Capella x 0.08 5.28 46.00 81.86 22.86 5 16 41.4 + 45 59 535 Bet Ori Rigel x 0.12 5.24 -8.20 76.83 -31.12 5 14 32.3 - 8 12 66 Alp CMi Procyon x 0.38 7.66 5.00 115.83 -16.24 7 39 18.1 + 5 13 307 Alp Ori Betelgeuse x 0.50 5.92 7.41 88.75 -16.03 5 55 10.3 + 7 24 258 Alp Aql Altair x 0.77 19.85 8.87 301.78 29.30 19 50 47.0 + 8 52 69 Alp Tau Aldebaran x 0.85 4.60 16.51 69.79 -5.47 4 35 55.2 + 16 30 3310 Alp Sco Antares 0.96 16.49 -26.43 249.76 -4.57 16 29 24.4 - 26 25 5511 Alp Vir Spica x 0.98 13.42 -11.16 203.84 -2.05 13 25 11.6 - 11 9 4112 Bet Gem Pollux x 1.14 7.78 28.03 113.54 6.74 7 45 18.9 + 28 1 3413 Alp PsA Fomalhaut 1.16 22.96 -29.62 333.86 -21.14 22 57 39.1 - 29 37 2014 Alp Cyg Deneb x 1.25 20.69 45.28 335.33 59.91 20 41 25.9 + 45 16 4915 Alp Leo Regulus x 1.35 10.14 11.97 149.83 0.46 10 8 22.3 + 11 58 216 Eps CMa Adhara 1.50 6.98 -28.97 110.76 -51.36 6 58 37.5 - 28 58 2017 Bet Tau Elnath 1.65 5.44 28.61 82.57 5.39 5 26 17.5 + 28 36 2718 Eps Ori Alnilam 1.70 5.60 -1.20 83.46 -24.51 5 36 12.8 - 1 12 719 Eps UMa Alioth 1.77 12.90 55.96 158.93 54.32 12 54 1.7 + 55 57 3520 Alp UMa Dubhe x 1.79 11.06 61.75 135.20 49.68 11 3 43.7 + 61 45 321 Alp Per Mirfak (Algenib) x 1.79 3.41 49.86 62.08 30.13 3 24 19.4 + 49 51 4022 Eta UMa Alkaid x 1.86 13.79 49.31 176.93 54.39 13 47 32.4 + 49 18 4823 Gam Gem Alhena 1.93 6.63 16.40 99.10 -6.74 6 37 42.7 + 16 23 5724 Alp Hya Alphard x 1.98 9.46 -8.66 147.28 -22.38 9 27 35.2 - 8 39 3125 Alp Ari Hamal x 2.00 2.12 23.46 37.66 9.96 2 7 10.4 + 23 27 4526 Alp UMi Polaris x 2.02 2.53 89.26 88.57 66.10 2 31 48.7 + 89 15 5127 Sig Sgr Nunki 2.02 18.92 -26.30 282.39 -3.45 18 55 15.9 - 26 17 4828 Bet Cet Diphda 2.04 0.73 -17.99 2.58 -20.78 0 43 35.4 - 17 59 1229 Bet And Mirach 2.06 1.16 35.62 30.41 25.94 1 9 43.9 + 35 37 1430 Alp And Alpheratz (Sirah) x 2.06 0.14 29.09 14.31 25.68 0 8 23.3 + 29 5 2631 Kap Ori Saiph 2.06 5.80 -9.67 86.40 -33.07 5 47 45.4 - 9 40 1132 Bet UMi Kochab 2.08 14.85 74.16 133.32 72.99 14 50 42.3 + 74 9 2033 Alp Oph Rasalhague x 2.08 17.58 12.56 262.45 35.84 17 34 56.1 + 12 33 3634 Bet Per Algol 2.12 3.14 40.96 56.17 22.43 3 8 10.1 + 40 57 2035 Bet Leo Denebola 2.14 11.82 14.57 171.62 12.27 11 49 3.6 + 14 34 1936 Gam Cyg Sadr 2.20 20.37 40.26 324.84 57.12 20 22 13.7 + 40 15 2437 Alp Cas Schedar 2.23 0.68 56.54 37.78 46.62 0 40 30.5 + 56 32 1438 Gam Dra Eltanin (Etamin) 2.23 17.94 51.49 267.97 74.92 17 56 36.4 + 51 29 2039 Alp CrB Alphecca x 2.23 15.58 26.71 222.30 44.32 15 34 41.3 + 26 42 5340 Gam1 And Almaak 2.26 2.07 42.33 44.23 27.81 2 3 54.0 + 42 19 4741 Bet Cas Caph x 2.27 0.15 59.15 35.12 51.21 0 9 10.7 + 59 8 5942 Zet UMa Mizar 2.27 13.40 54.93 165.70 56.38 13 23 55.5 + 54 55 3143 Del Sco Dschubba 2.32 16.01 -22.62 242.57 -1.99 16 0 20.0 - 22 37 1844 Bet UMa Merak 2.37 11.03 56.38 139.43 45.13 11 1 50.5 + 56 22 5745 Eps Peg Enif x 2.39 21.74 9.88 331.88 22.10 21 44 11.2 + 9 52 3046 Bet Peg Scheat x 2.42 23.06 28.08 359.37 31.14 23 3 46.5 + 28 4 5847 Eta Oph Sabik x 2.43 17.17 -15.72 257.97 7.20 17 10 22.7 - 15 43 2948 Alp Cep Alderamin 2.44 21.31 62.59 12.78 68.91 21 18 34.8 + 62 35 849 Gam UMa Phad 2.44 11.90 53.69 150.48 47.14 11 53 49.8 + 53 41 4150 Eps Cyg Gienah Cygni 2.46 20.77 33.97 327.75 49.42 20 46 12.7 + 33 58 13
** sono le stelle che con maggiore frequenza sono rappresentate sulla rete di un astrolabio
34
Trigonometria piana
Definizioni
sin(x) = a / c csc(x) = 1 / sin(x) = c / a cos(x) = b / c sec(x) = 1 / cos(x) = c / b tan(x) = sin(x) / cos(x) = a/b cot(x) = 1 / tan(x) = b / a
π 3.14159265359 π / 180 = 1 rad 57.29577951308 180 / π 0.01745329252
Identità trigonometriche
sin(-x) = -sin(x) cos x = sin(π/2 x) csc(-x) = -csc(x) sin x = cos(π/2 x) cos(-x) = cos(x) cox x = tan(π/2 x) sec(-x) = sec(x) tan x = cot(π/2 x) tan(-x) = -tan(x) csc x = sec(π/2 x) cot(-x) = -cot(x) sec x = csc(π/2 x) sin^2(x) + cos^2(x) = 1 tan^2(x) + 1 = sec^2(x) cot^2(x) + 1 = csc^2(x) sin(x±y) = sin x cos y ± cos x sin y cos(x±y) = cos x cosy -+sin x sin y tan(x±y) = (tan x±tan y) / (1 -+ tan x tan y) sin(2x) = 2 sin x cos x cos(2x) = cos^2(x) - sin^2(x) = 2 cos^2(x) - 1 = 1 - 2 sin^2(x) tan(2x) = 2 tan(x) / (1 - tan^2(x))
Elementi di un triangolo
Triangolo abc con angoli A,B,C; a è opposto a A, b opposto a B, c opposto a C: a/sin(A) = b/sin(B) = c/sin(C) (Legge dei seni) c^2 = a^2 + b^2 - 2ab cos(C) b^2 = a^2 + c^2 - 2ac cos(B) (Legge dei coseni) a^2 = b^2 + c^2 - 2bc cos(A)
(a - b)/(a + b) = tan 1/2(A-B) / tan 1/2(A+B) (Legge delle tangenti)
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Funzioni di angoli opposti, complementari, supplementari
-x 90-x 90+x 180-x sin -sin cos cos sincos cos sin -sin -costan -tan cot -cot -tancot -cot tan -tan -cot
Valori principali Angolo sin cos tan
0 0 1 07.5 0.13 0.99 0.1315 0.26 0.97 0.27
22.5 0.38 0.92 0.4130 0.5 0.87 0.57
37.5 0.61 0.8 0.7645 0.71 0.71 1
52.5 0.8 0.61 1.3160 0.87 0.5 1.74
67.5 0.92 0.38 2.4275 0.97 0.26 3.73
82.5 0.99 0.13 7.6290 1 0 ∞
Calcolo dell'arcotangente L'arcotangente va calcolata nel quadrante giusto, meglio calcolare numeratore e denominatore separatamente con i loro segni, il quadrante corretto per il risultato sarà:
tan x = A/B A B x + + 0÷90 + - 90÷180 - + 270÷360 - - 180÷270
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Trigonometria sferica
Sfera trigonometrica Le principali definizioni di geometria sulla sfera possono essere date utilizzando una sfera di raggio unitario, detta sfera trigonometrica, con il centro nellorigine di una terna di assi cartesiani orientati come in Fig. 1. Di seguito sono riportate le principali definizioni e formule di maggior utilizzo nei calcoli astronomici e nautici. Circolo massimo: qualsiasi cerchio ottenuto tagliando la sfera con un piano passante per il suo centro. I circoli massimi ottenuti con i piani fondamentali xy, xz ed yz sono anche detti circoli massimi fondamentali. Circolo minore: qualsiasi cerchio ottenuto tagliando la sfera con un piano non passante per il centro. Distanza sferica fra due punti sulla sfera: coincide con larco di circolo massimo, minore di mezza circonferenza, che ha per estremi quei punti. La distanza viene normalmente misurata in gradi e corrisponde allangolo al centro sotteso dal suddetto arco. Poli o centri sferici di un circolo: sono gli estremi del diametro della sfera, perpendicolare al piano del circolo e passante per il centro della sfera.
Raggi sferici di un circolo: sono le distanze sferiche di un polo da un punto qualsiasi del circolo; normalmente dei due poli si considera quello a cui corrisponde un raggio sferico minore di mezza circonferenza.
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Angolo fra due circoli massimi: è langolo formato dalle tangenti in uno dei loro due punti comuni, coincidente anche con langolo diedro fra i due piani formanti i suddetti circoli. Se uno dei due circoli passa per i poli dellaltro, langolo fra i due è retto. Perpendicolare sferica: dato un circolo massimo C ed un punto P (non appartenente a C), la perpendicolare sferica coincide con il circolo massimo passante per P e per i poli del cerchio C. Larco fra il punto P ed il punto intersezione della perpendicolare con il cerchio C, rappresenta la distanza sferica del punto dal cerchio C. Triangolo sferico è la superficie sulla sfera trigonometrica limitata da tre archi di circolo massimo passanti per tre punti detti vertici; tali punti non devono appartenere allo stesso circolo massimo e gli archi non devono avere alcun punto d'intersezione al di fuori dei vertici. Si può facilmente verificare che tre punti su una sfera definiscono otto triangoli sferici, ma soltanto uno ha tutti i lati più piccoli di una semicirconferenza. Lati del triangolo sferico: sono le lunghezze degli archi AB, BC, CA che limitano la superficie. Tali lati, per quanto già detto, sono minori o uguali a 180°. Angoli del triangolo sferico: sono gli angoli formati dai tre archi di circolo massimo. Eccesso sferico: è dato dalla differenza ∆ = A+B+C-180° in quanto in un triangolo sferico la somma degli angoli può essere compresa fra 1 e 3 angoli piatti.
Triangoli sferici rettangoli: sono i triangoli con un angolo retto; possono esistere anche triangoli con due o tre angoli retti, chiamati triangoli birettangoli o trirettangoli. Triangoli sferici rettilateri: sono i triangoli con un lato lungo 90°. Analogamente possono esistere triangoli birettilateri e trirettilateri.
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Principali relazioni trigonometriche Formule di Eulero (Teorema del coseno) Il coseno di un lato è uguale al prodotto dei coseni degli altri due lati più il prodotto dei seni moltiplicati per il coseno dell'angolo opposto.
cos a = cos b cos c + sin b sin c cos A cos b = cos a cos c + sin a sin c cos B cos c = cos a cos b + sin a sin b cos B
Teorema dei seni Il rapporto fra il seno di un angolo ed il seno del lato opposto è costante.
sin A -------- sin a
=sin B
-------- sin b
=sin C
-------- sin c
Teorema delle cotangenti Sono formule che legano fra loro quattro elementi consecutivi: due lati e due angoli; complessivamente sono sei formule che si possono ricavare ricorrendo ad una regola mnemonica illustrata di seguito con un esempio. Siano a, C, b, A i quattro elementi consecutivi da legare con la suddetta formula. Si disegna il triangolo sferico e si traccia una linea spezzata come mostrato in figura, si parte dal lato esterno a (lato non compreso fra i due angoli), si va all'altro lato c, si ritorna nell'angolo compreso C e si raggiunge infine l'angolo A opposto al lato a di partenza.
Si scrivono due terne di funzioni, di cui la prima è Cot, Sin e Cos (cotangente, seno e coseno) e la seconda è limmagine speculare della prima; le sei funzioni trigonometriche vanno divise in tre coppie, fra la prima e la seconda si pone il segno d'uguaglianza, fra le ultime due si pone il segno più. Gli argomenti delle sei funzioni trigonometriche sono nell'ordine quelli indicati dalla precedente spezzata, con l'avvertenza di scrivere due volte gli elementi corrispondenti alle cuspidi della spezzata: angolo C e lato b nell'esempio di figura. Nellesempio proposto si ha in definitiva:
cot a sin b = cos b cos C + sin C cot A
Le altre relazioni sono: cot a sin c = cos c cos B + sin B cot A cot b sin a = cos a cos C + sin C cot B cot b sin c = cos c cos A + sin A cot B cot c sin a = cos a cos B + sin B cot C cot c sin b = cos b cos A + sin A cot C
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Triangoli sferici rettangoli Tutte le formule precedenti si semplificano notevolmente nel caso di triangoli sferici rettangoli; considerando anche altre formule non riportate precedentemente, si ottengono dieci formule ridotte che possono essere facilmente ricordate con la regola mnemonica di Nepero. Si disegna una stella a cinque punte ed in ogni settore si scrivono consecutivamente tutti gli elementi del triangolo saltando l'angolo retto e sostituendo i cateti con i loro complementi. Il coseno di un elemento è uguale al prodotto delle cotangenti degli elementi adiacenti oppure è uguale al prodotto dei seni degli elementi opposti.
Nel trascrivere gli elementi nei vari settori non importa da quale si parte e dal verso (orario o antiorario). In Fig. 4 è riportato il caso dellangolo retto in A, applicando la precedente regola al lato a si ottengono:
cos a = cot B cot C cos a = sin(90°-b) sin(90°-c) = cos b cos c
Procedendo analogamente per gli altri quattro elementi si ottengono in totale le dieci formule già menzionate.
Plane Trigonometry Spherical Trigonometry
valid if a,b,c <<<1 rad
Law of Sines
Law of Cosines
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Simboli GST Tempo siderale a Greenwich (ore) LST Tempo siderale locale (ore) HA angolo orario JD giorno giuliano d2000 giorni dal 1 gennaio 2000 0h UT dMM giorno del mese dAA giorno dell'anno = numero di giorni dal 1 gennaio T secoli giuliani (36525 giorni) dall'epoca J2000.0 L Longitudine geografica (est -; ovest + ) ϕ Latitudine geografica UT Tempo Universale (ora di Greenwich) TDT Tempo Dinamico Terrestre (~ET, tempo delle effemeridi) ∆T Delta T (UT-TDT) Et Equazione del tempo (Sole vero-Sole medio) AA anno (se anno d.C.: AA = AA, se anno a.C.: AA = -(AA - 1) MM mese GG giorno hh ora mm minuti ss secondi β latitudine celeste λ longitudine celeste ε obliquità dell'eclittica A azimuth h altezza δ declinazione α ascensione retta J2000.0 = 2451545.0 = 1 gen 2000 12h UT X mod Y = resto di X/Y int(X) = parte intera di X frac(X) = parte decimale di X X \ Y = divisione intera = parte intera di X/Y (ignorare i decimali) ore = gradi / 15 gradi = ore · 15 PosF(X) = frac(X) se X>0, 1-frac(X) se X<0
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Zodiaco Segno Abbrev λ° Simb
Ariete Aries Ari 0 " Toro Taurus Tau 30 $ Gemelli Gemini Gem 60 % Cancro Cancer Cnc 90 & Leone Leo Leo 120 ' Vergine Virgo Vir 150 ( Bilancia Libra Lib 180 ) Scorpione Scorpio Sco 210 * Sagittario Sagittarius Sgr 240 + Capricorno Capricornus Cap 270 , Acquario Aquarius Aqr 300 - Pesci Pisces Psc 330 .
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Alfabeto greco
alpha α Α nu ν Ν beta β Β omicron ο Ο gamma γ Γ pi π Π delta δ ∆ rho ρ Ρ epsilon ε Ε sigma σ,ς Σ zeta ζ Ζ tau τ Τ eta η Η upsilon υ Υ theta θ Θ phi ϕ Φ iota ι Ι chi χ Χ kappa κ Κ xi ξ Ξ lambda λ Λ psi ψ Ψ mu µ Μ omega ω Ω
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Costellazioni
Centro approssim. (J2000.0) Nome latino Nome italiano abbr. αh αm δ δ° δ' Andromeda Andromeda And 0 34 + 39 15Antlia Macchina Pneumatica Ant 10 7 - 33 21Apus Uccello del Paradiso Aps 16 8 - 76 35Aquarius Acquario Aqr 22 42 - 10 28Aquila Aquila Aql 19 41 + 3 22Ara Altare Ara 17 14 - 51 7Aries Ariete Ari 2 41 + 22 34Auriga Auriga Aur 5 57 + 42 49Boötes Boote Boo 14 41 + 32 20Caelum Bulino Cae 4 43 - 38 10Camelopardus Giraffa Cam 6 9 + 71 58Cancer Cancro Cnc 8 30 + 2 3 3 4Canes Venatici Cani da Caccia CVn 13 1 + 42 21Canis Major Cane Maggiore CMa 6 50 - 22 19Canis Minor Cane Minore CMi 7 37 + 6 46Capricornus Capricorno Cap 21 3 - 19 21Carina Carena Car 7 46 - 57 50Cassiopeia Cassiopea Cas 0 52 + 60 18Centaurus Centa uro Cen 12 57 - 44 0Cepheus Cefeo Cep 22 25 + 72 34Cetus Balena Cet 1 43 - 6 22Chamaeleon Camal eonte Cha 12 0 - 81 1Circinus Compasso Cir 14 32 - 67 18Columba Colomba Col 5 42 - 37 55Coma Berenices Chioma di Berenice Com 12 45 + 22 39Corona Australis Corona Australe CrA 18 39 - 41 45Corona Borealis Corona Boreale CrB 15 53 + 32 38Corvus Corvo Crv 12 23 - 18 38Crater Cratere Crt 11 21 - 13 45Crux Croce del Sud Cru 12 29 - 60 18Cygnus Cigno Cyg 20 36 + 49 35Delphinus Delfino Del 20 40 + 12 6Dorado Dorado Dor 5 20 - 63 1Draco Dra gone Dra 17 57 + 66 4Equuleus Cavallu ccio Equ 21 15 + 7 56Eridanus Eridano Eri 3 53 - 17 59Fornax Forn ace For 2 46 - 26 4Gemini Gemelli Gem 6 51 + 24 49Grus Gru Gru 22 27 - 45 8Hercules Ercole Her 17 26 + 31 14Horologium Orolo gio Hor 3 13 - 52 0Hydra Idra Hya 9 8 - 11 41Hydrus Idra Australe Hyi 2 35 - 72 55Indus Indiano Ind 21 8 - 52 19Lacerta Lucertola Lac 22 31 + 46 40Leo Leone Leo 10 0 + 7 0Leo Minor Leone Minore LMi 10 19 + 33 14Lepus Lepre Lep 5 26 - 19 39Libra Bilancia Lib 15 11 - 15 33
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Centro approssim. (J2000.0) Nome latino Nome italiano abbr. αh αm δ δ° δ' Lupus Lupo Lup 15 23 - 42 43Lynx Lince Lyn 7 44 + 47 50Lyra Lira Lyr 18 54 + 40 39Mensa Mensa Men 5 30 - 79 1Microscopium Micr oscopio Mic 20 57 - 36 48Monoceros Uni corno Mon 6 58 - 3 16Musca Mosca Mus 12 28 - 69 8Norma Squadra Nor 16 3 - 52 43Octans Ottante Oct 22 10 - 84 16Ophiuchus Ofiuco Oph 17 2 - 2 21Orion Orion e Ori 5 34 + 3 35Pavo Pavone Pav 19 10 - 65 52Pegasus Pegaso Peg 22 37 + 19 39Perseus Perseo Per 3 31 + 44 53Phoenix Fenice Phe 0 44 - 48 46Pictor Pittore Pic 5 23 - 51 22Pisces Pesci Psc 0 53 + 15 29Piscis Austrinus Pesce Australe PsA 22 25 - 31 34Puppis Poppa Pup 7 52 - 32 37Pyxis Bussola Pyx 8 53 - 29 47Reticulum Reticolo Ret 3 54 - 60 31Sagitta Saetta Sge 19 40 + 17 0Sagittarius Sagittario Sgr 19 23 - 29 53Scorpius Scorpione Sco 16 52 - 35 20Sculptor Scultore Scl 1 0 - 38 31Scutum Scudo Sct 18 39 - 10 53Serpens Caput Serpente Ser 15 44 + 10 51Serpens Cauda Serpente Ser 17 57 - 13 29Sextans Sestante Sex 10 6 - 1 8Taurus Toro Tau 4 6 + 17 20Telescopium Telescopio Tel 19 15 - 51 28Triangulum Triangolo Tri 2 3 + 32 20Triangulum Australe Triangolo Australe TrA 16 7 - 65 6Tucana Tucano Tuc 23 50 - 64 56Ursa Major Orsa Maggiore UMa 10 16 + 57 29Ursa Minor Orsa Minore UMi 14 58 + 75 2Vela Vela Vel 9 20 - 48 29Virgo Vergin e Vir 13 21 - 3 31Volans Pesce Volante Vol 7 40 - 69 37Vulpecula Volpetta Vul 20 22 + 25 2
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Costellazioni obsolete
Costellazione Inventore Posizione odierna stelle Argo a.C. (Ipparco, Tolomeo) Vela, Puppis e Carina Anser Heveliu s 1690 Vulpecula Antinous a.C. Aquila Cancer Minor Lubienitzki (1623-1675) Cancer e Gemini Caput Medusae Plinio e Ipparco Perseus Cor Caroli Halley 1677 Alpha Canes Venatici (stella
doppia) Cerberus Heveliu s 1687 Hercules Custos Messium o Le Messier de La Lande (1732-1807) Cepheus e Cassiopeia Felis de La Lande Hydra Fleur de Lys o Lilium Pardies 1674 e Royer 1679 Aries Gallus Plancius 1598 o Bartsch 1624 Argo Navis Globus Aerostaticus de La Lande 1798 Piscis Austrinus e Microscopium Honeres Frederici o Frederici Gloria
Bode 1787 Andromeda
Jordanus Fluvius Plancius 1613 Lochium Funis Bode 1787 Argo Navis Machina Electrica Bode 1787 Fornax Mons Maenalus Hevelius 1690, Flamsteed Bootes Musca Borealis (Apes, Vespa) Plancius 1598, Bartsch 1624 Aries Noctua Burritt 1833 Hydra, Libra e Virgo Norma Nilotica Burritt 1833 Aquarius Nubecula o Nubes Major Magellan 1519 Dorado e Mensa Nubecula o Nubes Minor Magellan 1519 Tucana Officina Typographica Bode 1787 Puppis Psalterium Georgii o Georgianum
Hell 1781, (Bode la chiama Harpa Georgii)
Eridanus
Quadrans Muralis de La Lande Bootes Quadratum Allard 1700 vicino alla Grande Nube di
Magellano Robur Carolinum Halley 1677 Argo Navis Sagitta Australis Hevelius 1645 Scorpius e Sagittarius Sceptre e Hand of Justice Roget 1679 Lacerta Sceptrum Brandenburgicum Kirch 1688 Eridanus Solarium Burritt 1833 Hydrus e Dorado Solitarius o Turdus Solitarius Le Monnier 1776 Hydra, Libra e Virgo Taurus Poniatowski Poczobut 1777 Ophiuchus Tarandus vel Rangifer Le Monnier 1776 Cepheus e Cassiopeia Telescopium Herscelii Major Hell 1781 Auriga Telescopium Herscelii Minor Hell 1781 Taurus Telescopium Herscelii Bode (combinando i precedenti)
1787 Taurus
Tigris o Euphrates Fluvius Bartsch 1624 Ophiuchus, Lyra, Cygnus PegasusTriangulum Minor Hevelius 1687 tre stelle a sud del Triangulum
Major verso Hamal (Alpha Arietis)
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Tavole trigonometriche gradi sin cos tan cotan radianti
0.00 0.000 00 1.000 00 0.00000 0.00000 0.30 0.008 73 0.999 96 0.00873 114.58865 0.00873 1.00 0.017 45 0.999 85 0.01746 57.28996 0.01745 1.30 0.026 18 0.999 66 0.02619 38.18846 0.02618 2.00 0.034 90 0.999 39 0.03492 28.63625 0.03491 2.30 0.043 62 0.999 05 0.04366 22.90377 0.04363 3.00 0.052 34 0.998 63 0.05241 19.08114 0.05236 3.30 0.061 05 0.998 13 0.06116 16.34986 0.06109 4.00 0.069 76 0.997 56 0.06993 14.30067 0.06981 4.30 0.078 46 0.996 92 0.07870 12.70620 0.07854 5.00 0.087 16 0.996 19 0.08749 11.43005 0.08727 5.30 0.095 85 0.995 40 0.09629 10.38540 0.09599 6.00 0.104 53 0.994 52 0.10510 9.51436 0.10472 6.30 0.113 20 0.993 57 0.11394 8.77689 0.11345 7.00 0.121 87 0.992 55 0.12278 8.14435 0.12217 7.30 0.130 53 0.991 44 0.13165 7.59575 0.13090 8.00 0.139 17 0.990 27 0.14054 7.11537 0.13963 8.30 0.147 81 0.989 02 0.14945 6.69116 0.14835 9.00 0.156 43 0.987 69 0.15838 6.31375 0.15708 9.30 0.165 05 0.986 29 0.16734 5.97576 0.16581
10.00 0.173 65 0.984 81 0.17633 5.67128 0.17453 10.30 0.182 24 0.983 25 0.18534 5.39552 0.18326 11.00 0.190 81 0.981 63 0.19438 5.14455 0.19199 11.30 0.199 37 0.979 92 0.20345 4.91516 0.20071 12.00 0.207 91 0.978 15 0.21256 4.70463 0.20944 12.30 0.216 44 0.976 30 0.22169 4.51071 0.21817 13.00 0.224 95 0.974 37 0.23087 4.33148 0.22689 13.30 0.233 45 0.972 37 0.24008 4.16530 0.23562 14.00 0.241 92 0.970 30 0.24933 4.01078 0.24435 14.30 0.250 38 0.968 15 0.25862 3.86671 0.25307 15.00 0.258 82 0.965 93 0.26795 3.73205 0.26180 15.30 0.267 24 0.963 63 0.27732 3.60588 0.27053 16.00 0.275 64 0.961 26 0.28675 3.48741 0.27925 16.30 0.284 02 0.958 82 0.29621 3.37594 0.28798 17.00 0.292 37 0.956 30 0.30573 3.27085 0.29671 17.30 0.300 71 0.953 72 0.31530 3.17159 0.30543 18.00 0.309 02 0.951 06 0.32492 3.07768 0.31416 18.30 0.317 30 0.948 32 0.33460 2.98868 0.32289 19.00 0.325 57 0.945 52 0.34433 2.90421 0.33161 19.30 0.333 81 0.942 64 0.35412 2.82391 0.34034 20.00 0.342 02 0.939 69 0.36397 2.74748 0.34907 20.30 0.350 21 0.936 67 0.37388 2.67462 0.35779 21.00 0.358 37 0.933 58 0.38386 2.60509 0.36652 21.30 0.366 50 0.930 42 0.39391 2.53865 0.37525 22.00 0.374 61 0.927 18 0.40403 2.47509 0.38397 22.30 0.382 68 0.923 88 0.41421 2.41421 0.39270 23.00 0.390 73 0.920 50 0.42447 2.35585 0.40143 23.30 0.398 75 0.917 06 0.43481 2.29984 0.41015 24.00 0.406 74 0.913 55 0.44523 2.24604 0.41888 24.30 0.414 69 0.909 96 0.45573 2.19430 0.42761 25.00 0.422 62 0.906 31 0.46631 2.14451 0.43633 25.30 0.430 51 0.902 59 0.47698 2.09654 0.44506 26.00 0.438 37 0.898 79 0.48773 2.05030 0.45379 26.30 0.446 20 0.894 93 0.49858 2.00569 0.46251 27.00 0.453 99 0.891 01 0.50953 1.96261 0.47124 27.30 0.461 75 0.887 01 0.52057 1.92098 0.47997 28.00 0.469 47 0.882 95 0.53171 1.88073 0.48869 28.30 0.477 16 0.878 82 0.54296 1.84177 0.49742 29.00 0.484 81 0.874 62 0.55431 1.80405 0.50615 29.30 0.492 42 0.870 36 0.56577 1.76749 0.51487 30.00 0.500 00 0.866 03 0.57735 1.73205 0.52360
47
gradi sin cos tan cotan radianti 30.00 0.500 00 0.86603 0.57735 1.73205 0.52360 30.30 0.507 54 0.86163 0.58905 1.69766 0.53233 31.00 0.515 04 0.85717 0.60086 1.66428 0.54105 31.30 0.522 50 0.85264 0.61280 1.63185 0.54978 32.00 0.529 92 0.84805 0.62487 1.60033 0.55851 32.30 0.537 30 0.84339 0.63707 1.56969 0.56723 33.00 0.544 64 0.83867 0.64941 1.53986 0.57596 33.30 0.551 94 0.83389 0.66189 1.51084 0.58469 34.00 0.559 19 0.82904 0.67451 1.48256 0.59341 34.30 0.566 41 0.82413 0.68728 1.45501 0.60214 35.00 0.573 58 0.81915 0.70021 1.42815 0.61087 35.30 0.580 70 0.81412 0.71329 1.40195 0.61959 36.00 0.587 79 0.80902 0.72654 1.37638 0.62832 36.30 0.594 82 0.80386 0.73996 1.35142 0.63705 37.00 0.601 82 0.79864 0.75355 1.32704 0.64577 37.30 0.608 76 0.79335 0.76733 1.30323 0.65450 38.00 0.615 66 0.78801 0.78129 1.27994 0.66323 38.30 0.622 51 0.78261 0.79544 1.25717 0.67195 39.00 0.629 32 0.77715 0.80978 1.23490 0.68068 39.30 0.636 08 0.77162 0.82434 1.21310 0.68941 40.00 0.642 79 0.76604 0.83910 1.19175 0.69813 40.30 0.649 45 0.76041 0.85408 1.17085 0.70686 41.00 0.656 06 0.75471 0.86929 1.15037 0.71558 41.30 0.662 62 0.74896 0.88473 1.13029 0.72431 42.00 0.669 13 0.74314 0.90040 1.11061 0.73304 42.30 0.675 59 0.73728 0.91633 1.09131 0.74176 43.00 0.682 00 0.73135 0.93252 1.07237 0.75049 43.30 0.688 35 0.72537 0.94896 1.05378 0.75922 44.00 0.694 66 0.71934 0.96569 1.03553 0.76794 44.30 0.700 91 0.71325 0.98270 1.01761 0.77667 45.00 0.707 11 0.70711 1.00000 1.00000 0.78540 45.30 0.713 25 0.70091 1.01761 0.98270 0.79412 46.00 0.719 34 0.69466 1.03553 0.96569 0.80285 46.30 0.725 37 0.68835 1.05378 0.94896 0.81158 47.00 0.731 35 0.68200 1.07237 0.93252 0.82030 47.30 0.737 28 0.67559 1.09131 0.91633 0.82903 48.00 0.743 14 0.66913 1.11061 0.90040 0.83776 48.30 0.748 96 0.66262 1.13029 0.88473 0.84648 49.00 0.754 71 0.65606 1.15037 0.86929 0.85521 49.30 0.760 41 0.64945 1.17085 0.85408 0.86394 50.00 0.766 04 0.64279 1.19175 0.83910 0.87266 50.30 0.771 62 0.63608 1.21310 0.82434 0.88139 51.00 0.777 15 0.62932 1.23490 0.80978 0.89012 51.30 0.782 61 0.62251 1.25717 0.79544 0.89884 52.00 0.788 01 0.61566 1.27994 0.78129 0.90757 52.30 0.793 35 0.60876 1.30323 0.76733 0.91630 53.00 0.798 64 0.60182 1.32704 0.75355 0.92502 53.30 0.803 86 0.59482 1.35142 0.73996 0.93375 54.00 0.809 02 0.58779 1.37638 0.72654 0.94248 54.30 0.814 12 0.58070 1.40195 0.71329 0.95120 55.00 0.819 15 0.57358 1.42815 0.70021 0.95993 55.30 0.824 13 0.56641 1.45501 0.68728 0.96866 56.00 0.829 04 0.55919 1.48256 0.67451 0.97738 56.30 0.833 89 0.55194 1.51084 0.66189 0.98611 57.00 0.838 67 0.54464 1.53986 0.64941 0.99484 57.30 0.843 39 0.53730 1.56969 0.63707 1.00356 58.00 0.848 05 0.52992 1.60033 0.62487 1.01229 58.30 0.852 64 0.52250 1.63185 0.61280 1.02102 59.00 0.857 17 0.51504 1.66428 0.60086 1.02974 59.30 0.861 63 0.50754 1.69766 0.58905 1.03847 60.00 0.866 03 0.50000 1.73205 0.57735 1.04720
48
gradi sin cos tan cotan radianti 60.00 0.866 03 0.50000 1.73205 0.57735 1.04720 60.30 0.870 36 0.49242 1.76749 0.56577 1.05592 61.00 0.874 62 0.48481 1.80405 0.55431 1.06465 61.30 0.878 82 0.47716 1.84177 0.54296 1.07338 62.00 0.882 95 0.46947 1.88073 0.53171 1.08210 62.30 0.887 01 0.46175 1.92098 0.52057 1.09083 63.00 0.891 01 0.45399 1.96261 0.50953 1.09956 63.30 0.894 93 0.44620 2.00569 0.49858 1.10828 64.00 0.898 79 0.43837 2.05030 0.48773 1.11701 64.30 0.902 59 0.43051 2.09654 0.47698 1.12574 65.00 0.906 31 0.42262 2.14451 0.46631 1.13446 65.30 0.909 96 0.41469 2.19430 0.45573 1.14319 66.00 0.913 55 0.40674 2.24604 0.44523 1.15192 66.30 0.917 06 0.39875 2.29984 0.43481 1.16064 67.00 0.920 50 0.39073 2.35585 0.42447 1.16937 67.30 0.923 88 0.38268 2.41421 0.41421 1.17810 68.00 0.927 18 0.37461 2.47509 0.40403 1.18682 68.30 0.930 42 0.36650 2.53865 0.39391 1.19555 69.00 0.933 58 0.35837 2.60509 0.38386 1.20428 69.30 0.936 67 0.35021 2.67462 0.37388 1.21300 70.00 0.939 69 0.34202 2.74748 0.36397 1.22173 70.30 0.942 64 0.33381 2.82391 0.35412 1.23046 71.00 0.945 52 0.32557 2.90421 0.34433 1.23918 71.30 0.948 32 0.31730 2.98868 0.33460 1.24791 72.00 0.951 06 0.30902 3.07768 0.32492 1.25664 72.30 0.953 72 0.30071 3.17159 0.31530 1.26536 73.00 0.956 30 0.29237 3.27085 0.30573 1.27409 73.30 0.958 82 0.28402 3.37594 0.29621 1.28282 74.00 0.961 26 0.27564 3.48741 0.28675 1.29154 74.30 0.963 63 0.26724 3.60588 0.27732 1.30027 75.00 0.965 93 0.25882 3.73205 0.26795 1.30900 75.30 0.968 15 0.25038 3.86671 0.25862 1.31772 76.00 0.970 30 0.24192 4.01078 0.24933 1.32645 76.30 0.972 37 0.23345 4.16530 0.24008 1.33518 77.00 0.974 37 0.22495 4.33148 0.23087 1.34390 77.30 0.976 30 0.21644 4.51071 0.22169 1.35263 78.00 0.978 15 0.20791 4.70463 0.21256 1.36136 78.30 0.979 92 0.19937 4.91516 0.20345 1.37008 79.00 0.981 63 0.19081 5.14455 0.19438 1.37881 79.30 0.983 25 0.18224 5.39552 0.18534 1.38754 80.00 0.984 81 0.17365 5.67128 0.17633 1.39626 80.30 0.986 29 0.16505 5.97576 0.16734 1.40499 81.00 0.987 69 0.15643 6.31375 0.15838 1.41372 81.30 0.989 02 0.14781 6.69116 0.14945 1.42244 82.00 0.990 27 0.13917 7.11537 0.14054 1.43117 82.30 0.991 44 0.13053 7.59575 0.13165 1.43990 83.00 0.992 55 0.12187 8.14435 0.12278 1.44862 83.30 0.993 57 0.11320 8.77689 0.11394 1.45735 84.00 0.994 52 0.10453 9.51436 0.10510 1.46608 84.30 0.995 40 0.09585 10.38540 0.09629 1.47480 85.00 0.996 19 0.08716 11.43005 0.08749 1.48353 85.30 0.996 92 0.07846 12.70620 0.07870 1.49226 86.00 0.997 56 0.06976 14.30067 0.06993 1.50098 86.30 0.998 13 0.06105 16.34986 0.06116 1.50971 87.00 0.998 63 0.05234 19.08114 0.05241 1.51844 87.30 0.999 05 0.04362 22.90377 0.04366 1.52716 88.00 0.999 39 0.03490 28.63625 0.03492 1.53589 88.30 0.999 66 0.02618 38.18846 0.02619 1.54462 89.00 0.999 85 0.01745 57.28996 0.01746 1.55334 89.30 0.999 96 0.00873 114.58865 0.00873 1.56207 90.00 1.000 00 0.00000 ∞ 0.00000 1.57080
49
Bibliografia Meeus, Jean, Astronomical Algorithms, Willmann-Bell 1a ed. inglese, 1991 Duffett-Smith, Peter, Practical Astronomy with your calculator, Cambridge University Press 3^ ed. 1988 Hawkins, G. S., Mindsteps to the Cosmos (1983). trad. it. Cinque passi verso il Cosmo, di Giuseppe Politi, Frassinelli, Milano 1984 Montenbruck, Pfleger, Astronomy on the personal computer, Springer 3rd edition 1988 Seidelmann, P. Kenneth (ed.), Explanatory Supplement to the Astronomical Almanac, University Science Books 1992
50
Carte e Mappe
51
52
53
54
55
56
57
58
Catalogo Storico
In questo catalogo sono riportate le coordinate equatoriali (AR e Declinazione) ed eclittiche (Longitudine e Latitudine) delle 173 stelle più brillanti (fino a mag 3.0) ridotte per l'equinozio medio dal 4713 a.C. ad oggi (AD 2000) ogni 500 anni.
4713
a.C
.45
00 a
.C.
4000
a.C
.#
BS
CId
Co
stN
om
eM
agA
R (
h)
Dec
(º)
Lo
n (
º)L
at (
º)A
R (
h)
Dec
(º)
Lo
n (
º)L
at (
º)A
R (
h)
Dec
(º)
Lo
n (
º)L
at (
º)1
2491
αC
Ma
Siri
us-1
.46
1.87
130
-29.
6484
712
.553
90-3
8.22
038
2.02
616
-28.
6636
415
.434
45-3
8.26
175
2.38
759
-26.
4549
222
.200
21-3
8.35
923
223
26α
Car
Can
opus
-0.7
24.
2199
0-6
0.20
416
12.6
3009
-76.
7724
04.
2705
1-5
9.66
500
15.5
6198
-76.
7410
74.
3970
1-5
8.45
922
22.4
4059
-76.
6675
03
5340
αB
ooA
rctu
rus
-0.0
48.
2119
756
.725
4811
1.53
056
35.1
0050
8.48
145
55.9
0306
114.
4450
834
.970
499.
0816
253
.725
3812
1.29
154
34.6
6292
454
59α
1C
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Ken
taur
us-0
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9.19
610
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4847
915
1.78
092
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6251
39.
3373
6-2
8.37
239
154.
5424
9-4
1.65
430
9.67
012
-30.
5590
616
1.02
846
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7235
25
7001
αLy
rV
ega
0.03
14.9
7881
50.0
4422
192.
1667
262
.153
4415
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7749
.199
4019
5.10
630
62.1
3879
15.3
4787
47.3
1995
202.
0084
162
.104
366
1708
αA
urC
apel
la0.
0822
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6316
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9334
9.44
479
22.8
1789
22.8
9978
17.6
7773
352.
3426
322
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9923
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0420
.388
9235
9.15
171
22.8
1584
717
13β
Ori
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120.
0200
7-3
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713
344.
0945
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1.94
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0.19
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0947
734
7.01
083
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9233
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5953
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1.26
615
353.
8610
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1.86
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829
43α
CM
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8652
7-4
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4.79
758
2.03
854
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3267
27.2
0595
-14.
8333
02.
4478
1-0
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3333
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89-1
4.91
765
947
2α
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Ach
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79-7
8.96
914
251.
4123
2-5
8.85
744
14.4
5010
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9826
625
4.35
479
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8817
115
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60-8
1.96
712
261.
2698
8-5
8.93
683
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136
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9292
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4153
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5.77
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0516
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6.90
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0.82
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9613
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8877
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6.83
644
1152
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Cen
Had
ar0.
618.
6424
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6.40
892
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9371
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3.47
453
8.78
739
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2015
314
4.81
525
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4903
29.
1280
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9.19
328
151.
5783
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3.52
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1275
57α
Aql
Alta
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7714
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5716
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0920
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501
29.6
7721
14.6
3165
15.5
3497
211.
1459
229
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5015
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0213
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1921
8.05
434
29.6
2528
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57α
Tau
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66-1
4.54
539
337.
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7922
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02-1
3.39
789
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0.64
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1.13
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1.17
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1.35
120
15.4
1915
24.5
7066
6.08
940
1.78
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8031
31.3
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0.41
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3.61
963
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1132
816
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2.57
291
250.
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4617
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5937
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6.97
607
60.4
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2.37
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2.26
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22-9
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5.04
189
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8931
211
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2.56
768
181.
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878
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6.75
457
358.
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Tau
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0.39
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5860
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2.98
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4.86
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1.82
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7.08
640
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7618
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5.54
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8343
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0.62
794
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1.85
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306
CA
T.5
#B
SC
IdC
ost
No
me
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hard
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3.79
166
CA
T.6
#B
SC
IdC
ost
No
me
Mag
9391
1α
Cet
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2377
11.3
3349
15.2
0711
-61.
7156
824
4.87
391
-41.
8259
715
.919
03-6
3.43
056
251.
8410
2-4
1.94
583
16.7
3584
-54.
8607
625
7.23
139
-32.
1961
217
.421
67-5
5.53
000
264.
2058
8-3
2.26
506
1.71
870
-64.
0395
833
4.97
928
-64.
2145
71.
9795
0-6
1.56
972
342.
1182
9-6
4.24
208
21.7
1264
-62.
6701
830
2.65
969
-45.
3538
622
.308
36-6
0.25
972
309.
6712
8-4
5.40
388
3.30
082
22.4
4242
53.0
1722
3.99
876
3.79
142
24.1
0500
59.9
9245
4.05
080
19.4
8928
43.9
7843
309.
3613
764
.452
1119
.749
5845
.130
8331
6.24
964
64.4
1407
21.3
2072
-18.
3180
531
6.54
431
-2.5
1576
21.7
8400
-16.
1272
232
3.54
241
-2.6
0155
5.87
854
22.6
2398
88.3
1811
-0.8
6958
6.38
267
22.5
1361
95.3
0179
-0.8
2010
3.41
294
38.4
2907
58.7
0815
19.0
5722
3.96
422
40.0
1028
65.6
7771
19.1
1464
14.5
7416
-66.
6757
824
2.44
060
-48.
0346
815
.315
17-6
8.67
944
249.
3934
4-4
8.10
277
15.4
8165
-24.
5516
523
5.96
662
-5.4
1143
15.9
8086
-26.
1141
724
2.93
984
-5.4
7544
15.8
5091
-24.
2619
824
0.82
552
-3.9
7211
16.3
5314
-25.
5927
824
7.79
954
-4.0
3736
18.6
6574
-21.
6790
227
9.27
656
1.50
337
19.1
6272
-21.
0236
128
6.25
168
1.43
709
6.99
945
9.17
354
105.
2286
0-1
3.53
959
7.45
250
8.28
944
112.
1915
1-1
3.48
712
12.5
3906
41.0
4380
167.
5693
640
.125
4912
.933
8138
.318
3317
4.56
687
40.1
2105
21.0
8532
-7.6
9043
316.
4246
28.
6524
521
.525
97-5
.571
1132
3.39
496
8.61
512
2.49
238
51.4
3002
53.0
6521
34.4
7198
3.07
994
53.5
0639
60.0
2146
34.5
3014
6.62
545
-50.
0767
311
0.84
220
-72.
9007
16.
8322
8-5
0.61
472
117.
7269
1-7
2.85
334
22.3
2882
27.6
4556
348.
7829
135
.112
6622
.716
6930
.221
3935
5.71
317
35.1
0837
3.57
954
-15.
0267
246
.863
37-3
3.24
069
3.96
717
-13.
5086
153
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64-3
3.20
226
12.0
6950
-13.
7191
718
6.50
739
-12.
1479
412
.497
75-1
6.51
556
193.
4519
5-1
2.19
634
16.8
9108
-49.
2948
125
7.96
115
-26.
4852
217
.530
69-4
9.87
611
264.
9342
7-2
6.56
053
21.6
6827
-2.0
4673
326.
6178
111
.286
6522
.096
39-0
.319
7233
3.35
237
10.6
6175
6.21
866
25.4
7867
92.9
6246
2.00
825
6.73
219
25.1
3111
99.9
3869
2.06
995
9.28
617
25.9
9651
133.
7246
59.
6808
59.
7641
923
.774
1714
0.70
499
9.71
533
4.44
042
42.9
1900
71.8
6835
20.8
8002
5.03
281
43.8
2333
78.8
4131
20.9
4442
17.5
6227
-30.
2742
626
4.29
113
-6.9
0040
18.0
9681
-30.
4241
727
1.26
148
-6.9
9123
18.7
0728
13.2
2738
282.
8420
736
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6019
.090
1713
.863
3328
9.79
594
36.1
8558
1.67
154
32.5
5019
35.3
7825
20.5
4465
2.15
906
34.9
8722
42.3
5246
20.5
8065
5.13
090
20.6
9455
77.8
0846
-2.2
5820
5.62
742
21.1
4250
84.7
8469
-2.1
9567
11.7
4377
-19.
8369
118
4.72
130
-19.
6537
812
.168
75-2
2.61
972
191.
6654
2-1
9.67
348
12.8
6735
-20.
4932
420
0.04
555
-13.
7064
813
.315
36-2
3.17
167
207.
0183
7-1
3.74
275
3.13
355
46.0
5279
57.8
3649
27.2
4592
3.71
542
47.7
8750
64.8
0205
27.3
0169
CA
T.2
4
83
Catalogo di Yale
Sono qui riportate le coordinate (J2000.0) delle 200 stelle più brillanti tratte dallo Yale Bright Star Catalogue (online su: Astronomical Data Center - NASA, http://adc.gsfc.nasa.gov/adc-cgi/cat.pl?/catalogs/5/5050/
#H
Rid
Co
stN
om
em
agp
mA
Rp
mD
eP
aral
lxV
Rad
AR
Dec
Lo
nL
ath
ms
° '
" ar
csec
/yr
arcs
ec/y
rar
csec
km/s
(h.d
ecim
)(°
.dec
im)
(°.d
ecim
)(°
.dec
im)
124
91α
CM
aS
irius
-1.4
66
458.
9-
1642
58-0
.553
-1.2
050.
375
-86.
7524
7-1
6.71
611
104.
0815
7-3
9.60
525
223
26α
Car
Can
opus
-0.7
26
2357
.1-
5241
450.
022
0.02
10.
028
216.
3991
9-5
2.69
583
104.
9605
5-7
5.82
406
353
40α
Boo
Arc
turu
s-0
.04
1415
39.7
+19
1057
-1.0
93-1
.998
0.09
0-5
14.2
6103
19.1
8250
204.
2336
930
.736
364
5459
α1
Cen
Rig
el K
enta
urus
-0.0
114
3935
.9-
6050
7-3
.642
0.69
90.
751
-22
14.6
5997
-60.
8352
823
9.47
856
-42.
5960
15
7001
αLy
rV
ega
0.03
1836
56.3
+38
471
0.20
20.
286
0.12
3-1
418
.615
6438
.783
6128
5.31
612
61.7
3280
617
08α
Aur
Cap
ella
0.08
516
41.4
+45
5953
0.07
6-0
.425
0.07
330
5.27
817
45.9
9806
81.8
5805
22.8
6432
717
13β
Ori
Rig
el0.
125
1432
.3-
812
60.
000
-0.0
010.
013
215.
2423
1-8
.201
6776
.829
72-3
1.12
281
829
43α
CM
iP
rocy
on0.
387
3918
.1+
513
30-0
.710
-1.0
230.
288
-37.
6550
35.
2250
011
5.78
545
-16.
0196
29
472
αE
riA
cher
nar
0.46
137
42.9
-57
1412
0.09
5-0
.035
0.02
616
1.62
858
-57.
2366
734
5.31
157
-59.
3781
710
2061
αO
riB
etel
geus
e0.
505
5510
.3+
724
250.
026
0.00
90.
005
215.
9195
37.
4069
488
.754
58-1
6.02
713
1152
67β
Cen
Had
ar0.
6114
349
.4-
6022
23-0
.032
-0.0
190.
009
614
.063
72-6
0.37
306
233.
7922
5-4
4.13
768
1275
57α
Aql
Alta
ir0.
7719
5047
.0+
852
60.
538
0.38
60.
198
-26
19.8
4639
8.86
833
301.
7764
329
.303
4713
1457
αT
auA
ldeb
aran
0.85
435
55.2
+16
3033
0.06
3-0
.190
0.04
854
4.59
867
16.5
0917
69.7
8901
-5.4
6744
1461
34α
Sco
Ant
ares
0.96
1629
24.4
-26
2555
-0.0
10-0
.020
0.02
4-3
16.4
9011
-26.
4319
424
9.76
206
-4.5
6992
1550
56α
Vir
Spi
ca0.
9813
2511
.6-
119
41-0
.041
-0.0
280.
023
113
.419
89-1
1.16
139
203.
8414
6-2
.054
5216
2990
βG
emP
ollu
x1.
147
4518
.9+
281
34-0
.628
-0.0
460.
094
37.
7552
528
.026
1111
3.21
547
6.68
407
1787
28α
PsA
Fom
alha
ut1.
1622
5739
.1-
2937
200.
333
-0.1
650.
149
722
.960
86-2
9.62
222
333.
8604
5-2
1.13
568
1848
53β
Cru
Mim
osa
1.25
1247
43.2
-59
4119
-0.0
48-0
.014
1612
.795
33-5
9.68
861
221.
6457
2-4
8.63
873
1979
24α
Cyg
Den
eb1.
2520
4125
.9+
4516
490.
003
0.00
2-0
.006
-520
.690
5345
.280
2833
5.32
909
59.9
0612
2047
30α
1C
ruA
crux
1.33
1226
35.9
-63
557
-0.0
36-0
.012
0.00
8-1
112
.443
31-6
3.09
917
221.
8700
1-5
2.87
891
2154
60α
2C
enP
roxi
ma
1.33
1439
36.1
-60
508
-3.6
460.
700
0.75
1-2
114
.660
03-6
0.83
556
239.
4792
2-4
2.59
609
2239
82α
Leo
Reg
ulus
1.35
108
22.3
+11
582
-0.2
480.
006
0.04
56
10.1
3953
11.9
6722
149.
8290
90.
4648
423
2618
εC
Ma
Adh
ara
1.50
658
37.5
-28
5820
0.00
40.
003
0.00
127
6.97
708
-28.
9722
211
0.76
263
-51.
3603
824
4763
γC
ruG
acru
x1.
6312
319.
9-
576
480.
023
-0.2
6221
12.5
1942
-57.
1133
321
6.73
959
-47.
8313
725
6527
λS
coS
haul
a1.
6317
3336
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376
14-0
.001
-0.0
29-3
17.5
6014
-37.
1038
926
4.58
565
-13.
7885
226
1790
γO
riB
ella
trix
1.64
525
7.9
+6
2059
-0.0
09-0
.014
0.02
918
5.41
886
6.34
972
80.9
4662
-16.
8160
327
1791
βT
auE
lnat
h1.
655
2617
.5+
2836
270.
022
-0.1
750.
028
95.
4381
928
.607
5082
.574
895.
3851
628
3685
βC
arM
iapl
acid
us1.
689
1312
.0-
6943
2-0
.162
0.10
80.
021
-59.
2200
0-6
9.71
722
211.
9689
4-7
2.23
597
2919
03ε
Ori
Aln
ilam
1.70
536
12.8
-1
127
0.00
1-0
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-0.0
0226
5.60
356
-1.2
0194
83.4
6358
-24.
5064
130
8425
αG
ruA
lnai
r1.
7422
814
.0-
4657
400.
129
-0.1
510.
057
1222
.137
22-4
6.96
111
315.
9069
6-3
2.91
339
3149
05ε
UM
aA
lioth
1.77
1254
1.7
+55
5735
0.11
2-0
.006
0.00
9-9
12.9
0047
55.9
5972
158.
9334
454
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6432
3207
γ2
Vel
Reg
or1.
788
932
.0-
4720
12-0
.004
0.00
60.
017
358.
1588
9-4
7.33
667
147.
3499
6-6
4.46
417
3310
17α
Per
Mirf
ak1.
793
2419
.4+
4951
400.
024
-0.0
250.
016
-23.
4053
949
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1162
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0030
.125
3834
4301
αU
Ma
Dub
he1.
7911
343
.7+
6145
3-0
.119
-0.0
670.
038
-911
.062
1461
.750
8313
5.19
764
49.6
8017
3526
93δ
CM
aW
ezen
1.84
78
23.5
-26
2336
-0.0
030.
004
-0.0
1134
7.13
986
-26.
3933
311
3.39
615
-48.
4533
236
6879
εS
grK
aus
Aus
tral
is1.
8518
2410
.3-
3423
5-0
.038
-0.1
240.
023
-15
18.4
0286
-34.
3847
227
5.07
847
-11.
0519
037
3307
εC
arA
vior
1.86
822
30.8
-59
3035
-0.0
260.
014
28.
3752
2-5
9.50
972
173.
1296
5-7
2.68
011
3851
91η
UM
aA
lkai
d1.
8613
4732
.4+
4918
48-0
.122
-0.0
110.
035
-11
13.7
9233
49.3
1333
176.
9329
654
.387
9739
6553
θS
coS
arga
s1.
8717
3719
.2-
4259
520.
015
-0.0
020.
027
117
.622
00-4
2.99
778
265.
5996
4-1
9.64
506
4020
88β
Aur
Men
kalin
an1.
905
5931
.7+
4456
51-0
.057
0.00
00.
041
-18
5.99
214
44.9
4750
89.9
1030
21.5
0825
4162
17α
TrA
Atr
ia1.
9216
4839
.9-
691
400.
014
-0.0
340.
031
-316
.811
08-6
9.02
778
260.
8959
0-4
6.15
142
AR
D
ec
BS
C.1
#H
Rid
Co
stN
om
em
agp
mA
Rp
mD
eP
aral
lxV
Rad
AR
Dec
Lo
nL
ath
ms
° '
" ar
csec
/yr
arcs
ec/y
rar
csec
km/s
(h.d
ecim
)(°
.dec
im)
(°.d
ecim
)(°
.dec
im)
AR
D
ec
4224
21γ
Gem
Alh
ena
1.93
637
42.7
+16
2357
0.04
2-0
.042
0.03
3-1
36.
6285
316
.399
1799
.104
59-6
.742
6543
7790
αP
avP
eaco
ck1.
9420
2538
.9-
5644
60.
007
-0.0
892
20.4
2747
-56.
7350
029
3.81
762
-36.
2675
744
3485
δV
el1.
968
4442
.2-
5442
300.
023
-0.0
780.
051
28.
7450
6-5
4.70
833
168.
9469
0-6
7.19
747
4522
94β
CM
aM
irzam
1.98
622
42.0
-17
5721
-0.0
060.
000
0.01
934
6.37
833
-17.
9558
397
.188
02-4
1.25
363
4628
91α
Gem
Cas
tor
1.98
734
36.0
+31
5318
-0.1
71-0
.098
0.06
76
7.57
667
31.8
8833
110.
2406
510
.095
8847
3748
αH
yaA
lpha
rd1.
989
2735
.2-
839
31-0
.014
0.03
30.
022
-49.
4597
8-8
.658
6114
7.27
904
-22.
3824
648
617
αA
riH
amal
2.00
27
10.4
+23
2745
0.19
0-0
.148
0.04
9-1
42.
1195
623
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5037
.662
499.
9652
049
5958
TC
rB2.
0015
5930
.2+
2555
13-0
.005
0.01
3-2
915
.991
7225
.920
2823
0.10
028
45.2
7500
5042
4α
UM
iP
olar
is2.
022
3148
.7+
8915
510.
038
-0.0
150.
007
-17
2.53
019
89.2
6417
88.5
6767
66.1
0151
5171
21σ
Sgr
Nun
ki2.
0218
5515
.9-
2617
480.
013
-0.0
54-1
118
.921
08-2
6.29
667
282.
3852
4-3
.449
4552
188
βC
etD
iphd
a2.
040
4335
.4-
1759
120.
234
0.03
30.
061
130.
7265
0-1
7.98
667
2.58
359
-20.
7836
553
1948
ζO
riA
lnita
k2.
055
4045
.5-
156
340.
003
-0.0
020.
024
185.
6793
1-1
.942
7884
.681
19-2
5.29
330
5415
αA
ndA
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Copyright Questo documento è di proprietà di Rodolfo Baggio, © Copyright 2000-2001, tutti i diritti sono riservati. All'utente è garantita una licenza gratuita non esclusiva per l'uso a fini personali o di studio o per qualunque altro scopo ammesso dalla legge, purché NON commerciale. L'utilizzo a fini commerciali deve essere autorizzato esplicitamente dall'autore. Questo documento può essere distribuito e riprodotto liberamente purché non venga chiesto nessun compenso e le copie restino complete e non modificate. Questo documento è distribuito nello stato in cui si trova, nessuna garanzia, esplicita o implicita, viene data che il contenuto sia adatto a particolari scopi o che sia esente da errori od omissioni. L'utente si assume ogni responsabilità derivante dal possesso o dall'uso dei contenuti di questo documento. Milano, 24 settembre 2001 Rodolfo Baggio via Venini, 18 20127 Milano - Italy [email protected]