1322:
863:
502:
738:
1317:{\displaystyle {\begin{aligned}\phi _{s}&=\delta ,\\\lambda _{s}&=-15(T_{\mathrm {GMT} }-12+E_{\mathrm {min} }/60),\\S_{x}&=\cos \phi _{s}\sin(\lambda _{s}-\lambda _{o}),\\S_{y}&=\cos \phi _{o}\sin \phi _{s}-\sin \phi _{o}\cos \phi _{s}\cos(\lambda _{s}-\lambda _{o}),\\S_{z}&=\sin \phi _{o}\sin \phi _{s}+\cos \phi _{o}\cos \phi _{s}\cos(\lambda _{s}-\lambda _{o}).\end{aligned}}}
277:
497:{\displaystyle {\begin{aligned}\cos \phi _{\mathrm {s} }&={\frac {\sin \delta \cos \Phi -\cos h\cos \delta \sin \Phi }{\sin \theta _{\mathrm {s} }}}\\\cos \phi _{\mathrm {s} }&={\frac {\sin \delta -\cos \theta _{\mathrm {s} }\sin \Phi }{\sin \theta _{\mathrm {s} }\cos \Phi }}.\end{aligned}}}
2397:
Zhang, T., Stackhouse, P.W., Macpherson, B., and
Mikovitz, J.C., 2021. A solar azimuth formula that renders circumstantial treatment unnecessary without compromising mathematical rigor: Mathematical setup, application and extension of a formula based on the subsolar point and atan2 function.
258:
The following formulas can also be used to approximate the solar azimuth angle, but these formulas use cosine, so the azimuth angle as shown by a calculator will always be positive, and should be interpreted as the angle between zero and 180 degrees when the hour angle,
507:
So practically speaking, the compass azimuth which is the practical value used everywhere (in example in airlines as the so called course) on a compass (where North is 0 degrees, East is 90 degrees, South is 180 degrees and West is 270 degrees) can be calculated as
253:
98:
of south. For example, due east would be 90° and due west would be -90°. Another convention is the reverse; it also has the origin at due south, but measures angles clockwise, so that due east is now negative and west now positive.
567:
142:
The following formulas assume the north-clockwise convention. The solar azimuth angle can be calculated to a good approximation with the following formula, however angles should be interpreted with care because the
1815:
282:
2032:
1954:
868:
1675:
1599:
are the x-, y- and z-components, respectively, of the unit vector pointing toward the Sun. The x-, y- and z-axises of the coordinate system point to East, North and upward, respectively.
1731:
633:
602:
1480:
1443:
1597:
1542:
1406:
838:, that gives an unambiguous solution without the need for circumstantial treatment. The subsolar point is the point on the surface of the Earth where the Sun is overhead.
170:
1871:
1513:
1377:
2350:
1348:
687:
2117:
2090:
2063:
820:
793:
766:
712:
1842:
658:
2123:" as a graphic depiction of all possible positions of the Sun in terms of solar zenith angle and solar azimuth angle for any given location. Refer to
514:
2465:
2235:
2481:
2288:
2245:
849:
using equations from The
Astronomical Almanac, then it gives the x-, y- and z-components of the unit vector pointing toward the Sun, through
115:
2213:
2433:
2450:
1738:
2372:
1961:
1883:
127:
2454:
2437:
822:
at 1-hour step as viewed at the geographic center of the contiguous United States. The gray part indicates it is nighttime.
82:
There are several conventions for the solar azimuth; however, it is traditionally defined as the angle between a line due
2141:
43:
1606:
721:
In addition, dividing the above sine formula by the first cosine formula gives one the tangent formula as is used in
1683:
609:
854:
842:
578:
2491:
2344:
1450:
1413:
268:
2436:
with VBA functions for solar azimuth, solar elevation, dawn, sunrise, solar noon, sunset, and dusk, by
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2316:
2278:
2251:
248:{\displaystyle \sin \phi _{\mathrm {s} }={\frac {-\sin h\cos \delta }{\sin \theta _{\mathrm {s} }}}.}
2151:
690:
39:
1520:
1384:
1877:
If one prefers North-Clockwise
Convention, or East-Counterclockwise Convention, the formulas are
1849:
636:
59:
51:
1677:. With the above mathematical setup, the solar zenith angle and solar azimuth angle are simply
1491:
1355:
2459:
2364:
2332:
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2209:
741:"Wreath of Analemmas". The annual excursion of the position of the Sun defined by the triplet
103:
1333:
672:
2324:
2136:
1483:
846:
2445:
2307:
Reda, I., Andreas, A. (2004). "Solar
Position Algorithm for Solar Radiation Applications".
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2041:
798:
771:
744:
697:
850:
2441:
2320:
1827:
827:
643:
263:, is negative (morning) and the angle between 180 and 360 degrees when the hour angle,
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Free on-line tool to estimate the position of the sun with three different algorithms.
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2161:
826:
A 2021 publication presents a method that uses a solar azimuth formula based on the
2486:
737:
144:
107:
2328:
267:, is positive (afternoon). (These two formulas are equivalent if one assumes the "
130:
products, while also defining azimuthal angles relative to due north, take counter
114:, so east is 90°, south is 180°, and west is 270°. This is the definition used by
2468:
Azimuth Angle - online material regarding
Photovoltaics by UNSW, ASU, NSF et al.
102:
However, despite tradition, the most commonly accepted convention for analyzing
2399:
2386:
562:{\displaystyle {\text{compass }}\phi _{\mathrm {s} }=360-\phi _{\mathrm {s} }.}
2428:
2156:
2146:
2119:
at 1-hour step for an entire year can be presented in a 3D plot of "wreath of
835:
665:
661:
2336:
17:
131:
67:
118:
in their solar position calculators and is also the convention used in the
2410:
The
Astronomical Almanac for the Year. The United Naval Observatory, 2019.
2424:
Solar
Position Calculators by National Renewable Energy Laboratory (NREL)
2166:
2124:
2120:
715:
71:
2237:
Atmospheric
Chemistry and Physics, from Air Pollution to Climate Change
2171:
123:
55:
35:
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with a solar position and solar radiation time-series calculator, by
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1873:
is the solar azimuth angle following the South-Clockwise
Convention.
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736:
111:
87:
83:
63:
2429:
Solar
Position Algorithm for Solar Radiation Applications (NREL)
95:
91:
90:. This convention states the angle is positive if the shadow is
47:
164:, has multiple solutions, only one of which will be correct.
1810:{\displaystyle \gamma _{s}=\mathrm {atan2} (-S_{x},-S_{y})}
2206:
Solar Energy: Principles of Thermal Collection and Storage
2027:{\displaystyle \gamma _{s}=\mathrm {atan2} (S_{y},S_{x})}
1949:{\displaystyle \gamma _{s}=\mathrm {atan2} (S_{x},S_{y})}
2208:(3rd ed.). Tata McGraw-Hill Education. p. 84.
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247:
2440:, translated from NOAA's online calculators for
38:(horizontal angle with respect to north) of the
1670:{\displaystyle S_{x}^{2}+S_{y}^{2}+S_{z}^{2}=1}
2277:Duffie, John A.; Beckman, William A. (2013).
2234:Seinfeld, John H.; Pandis, Spyros N. (2006).
8:
2400:https://doi.org/10.1016/j.renene.2021.03.047
2387:https://thenauticalalmanac.com/Formulas.html
2349:: CS1 maint: multiple names: authors list (
2283:(4th ed.). Wiley. pp. 13, 15, 20.
572:The formulas use the following terminology:
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86:and the shadow cast by a vertical rod on
2272:
2270:
2268:
1726:{\displaystyle Z=\mathrm {acos} (S_{z})}
2398:Renewable Energy, 172, 1333-1340. DOI:
2193:
2127:for similar plots for other locations.
1408:is the longitude of the subsolar point,
2342:
2280:Solar Engineering of Thermal Processes
1379:is the latitude of the subsolar point,
628:{\displaystyle \theta _{\mathrm {s} }}
2302:
2300:
7:
2240:(2nd ed.). Wiley. p. 130.
2034:. (East-Counterclockwise Convention)
597:{\displaystyle \phi _{\mathrm {s} }}
110:applications, is clockwise from due
138:Conventional Trigonometric Formulas
27:Azimuth angle of the Sun's position
1988:
1985:
1982:
1979:
1910:
1907:
1904:
1901:
1765:
1762:
1759:
1756:
1703:
1700:
1697:
1694:
1475:{\displaystyle E_{\mathrm {min} }}
1466:
1463:
1460:
1445:is the Greenwich Mean Time or UTC,
1438:{\displaystyle T_{\mathrm {GMT} }}
1429:
1426:
1423:
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25:
1592:{\displaystyle S_{x},S_{y},S_{z}}
1544:is the longitude of the observer,
1515:is the latitude of the observer,
841:The method first calculates the
94:of south and negative if it is
2021:
1995:
1956:, (North-Clockwise Convention)
1943:
1917:
1817:. (South-Clockwise Convention)
1804:
1772:
1720:
1707:
1350:is the declination of the Sun,
1304:
1278:
1178:
1152:
1052:
1026:
977:
921:
1:
2329:10.1016/j.solener.2003.12.003
119:
70:) defines the Sun's apparent
2482:Horizontal coordinate system
2142:Horizontal coordinate system
1537:{\displaystyle \lambda _{o}}
1401:{\displaystyle \lambda _{s}}
78:Conventional sign and origin
1866:{\displaystyle \gamma _{s}}
2508:
1844:is the solar zenith angle,
604:is the solar azimuth angle
271:" approximation formula).
1508:{\displaystyle \phi _{o}}
1372:{\displaystyle \phi _{s}}
729:The formula based on the
122:presented here. However,
2204:Sukhatme, S. P. (2008).
834:function, as defined in
2460:Sun Position Calculator
2369:Landsat Data Dictionary
2038:Finally, the values of
1343:{\displaystyle \delta }
682:{\displaystyle \delta }
2113:
2086:
2059:
2028:
1950:
1867:
1838:
1811:
1727:
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1593:
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1402:
1373:
1344:
1318:
855:spherical trigonometry
843:declination of the Sun
823:
816:
789:
762:
733:and the atan2 function
708:
683:
654:
629:
598:
563:
498:
249:
2385:The Nautical Almanac
2114:
2112:{\displaystyle S_{z}}
2087:
2085:{\displaystyle S_{y}}
2060:
2058:{\displaystyle S_{x}}
2029:
1951:
1868:
1839:
1812:
1728:
1672:
1603:It can be shown that
1594:
1539:
1510:
1477:
1440:
1403:
1374:
1345:
1319:
817:
815:{\displaystyle S_{z}}
790:
788:{\displaystyle S_{y}}
763:
761:{\displaystyle S_{x}}
740:
709:
707:{\displaystyle \Phi }
684:
655:
630:
599:
564:
499:
269:solar elevation angle
250:
44:horizontal coordinate
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1962:
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723:The Nautical Almanac
698:
673:
644:
610:
579:
515:
278:
171:
134:angles as negative.
2321:2004SoEn...76..577R
2152:Position of the Sun
1660:
1642:
1624:
64:complementary angle
32:solar azimuth angle
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2082:
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637:solar zenith angle
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245:
60:solar zenith angle
52:relative direction
2451:An Excel workbook
2434:An Excel workbook
2290:978-0-470-87366-3
2247:978-0-471-72018-8
1837:{\displaystyle Z}
653:{\displaystyle h}
521:
485:
382:
240:
126:photos and other
104:solar irradiation
16:(Redirected from
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2250:. Archived from
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2137:Equation of time
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1484:equation of time
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847:equation of time
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851:vector analysis
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691:sun declination
689:is the current
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2455:Greg Pelletier
2448:
2446:sunrise/sunset
2442:solar position
2438:Greg Pelletier
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2419:
2418:External links
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2215:978-0070260641
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1658:
1653:
1649:
1645:
1640:
1635:
1631:
1627:
1622:
1617:
1613:
1601:
1600:
1586:
1582:
1578:
1573:
1569:
1565:
1560:
1556:
1545:
1531:
1527:
1516:
1502:
1498:
1487:
1468:
1465:
1462:
1457:
1446:
1431:
1428:
1425:
1420:
1409:
1395:
1391:
1380:
1366:
1362:
1351:
1339:
1325:
1324:
1309:
1306:
1301:
1297:
1293:
1288:
1284:
1280:
1277:
1274:
1269:
1265:
1261:
1258:
1253:
1249:
1245:
1242:
1239:
1234:
1230:
1226:
1223:
1218:
1214:
1210:
1207:
1204:
1201:
1199:
1195:
1191:
1187:
1186:
1183:
1180:
1175:
1171:
1167:
1162:
1158:
1154:
1151:
1148:
1143:
1139:
1135:
1132:
1127:
1123:
1119:
1116:
1113:
1108:
1104:
1100:
1097:
1092:
1088:
1084:
1081:
1078:
1075:
1073:
1069:
1065:
1061:
1060:
1057:
1054:
1049:
1045:
1041:
1036:
1032:
1028:
1025:
1022:
1017:
1013:
1009:
1006:
1003:
1000:
998:
994:
990:
986:
985:
982:
979:
976:
972:
965:
962:
959:
954:
950:
947:
944:
938:
935:
932:
927:
923:
920:
917:
914:
911:
909:
905:
901:
897:
896:
893:
890:
887:
884:
882:
878:
874:
870:
869:
857:, as follows:
828:subsolar point
809:
805:
782:
778:
755:
751:
734:
731:subsolar point
727:
719:
718:
703:
693:
678:
668:
649:
639:
621:
616:
605:
590:
585:
570:
569:
558:
552:
547:
543:
540:
537:
531:
526:
505:
504:
489:
483:
480:
477:
471:
466:
462:
459:
454:
451:
448:
442:
437:
433:
430:
427:
424:
421:
418:
412:
409:
407:
402:
397:
393:
390:
387:
386:
377:
372:
368:
365:
360:
357:
354:
351:
348:
345:
342:
339:
336:
333:
330:
327:
324:
321:
318:
315:
309:
306:
304:
299:
294:
290:
287:
284:
283:
256:
255:
244:
235:
230:
226:
223:
218:
215:
212:
209:
206:
203:
200:
194:
188:
183:
179:
176:
139:
136:
79:
76:
58:, whereas the
40:Sun's position
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2504:
2493:
2490:
2488:
2485:
2483:
2480:
2479:
2477:
2467:
2464:
2461:
2458:
2456:
2452:
2449:
2447:
2443:
2439:
2435:
2432:
2430:
2427:
2425:
2422:
2421:
2417:
2407:
2404:
2401:
2394:
2391:
2388:
2382:
2379:
2374:
2370:
2366:
2365:"Sun Azimuth"
2360:
2357:
2352:
2346:
2338:
2334:
2330:
2326:
2322:
2318:
2315:(5): 577–89.
2314:
2310:
2303:
2301:
2297:
2292:
2286:
2282:
2281:
2273:
2271:
2269:
2265:
2254:on 2013-09-06
2253:
2249:
2243:
2239:
2238:
2230:
2228:
2226:
2222:
2217:
2211:
2207:
2200:
2198:
2194:
2187:
2183:
2180:
2178:
2175:
2173:
2170:
2168:
2165:
2163:
2162:Solar tracker
2160:
2158:
2155:
2153:
2150:
2148:
2145:
2143:
2140:
2138:
2135:
2134:
2130:
2128:
2126:
2122:
2104:
2100:
2077:
2073:
2050:
2046:
2016:
2012:
2008:
2003:
1999:
1991:
1975:
1970:
1966:
1958:
1938:
1934:
1930:
1925:
1921:
1913:
1897:
1892:
1888:
1880:
1879:
1878:
1858:
1854:
1846:
1831:
1824:
1823:
1822:
1799:
1795:
1791:
1788:
1783:
1779:
1775:
1768:
1752:
1747:
1743:
1735:
1715:
1711:
1690:
1687:
1680:
1679:
1678:
1664:
1661:
1656:
1651:
1647:
1643:
1638:
1633:
1629:
1625:
1620:
1615:
1611:
1584:
1580:
1576:
1571:
1567:
1563:
1558:
1554:
1546:
1529:
1525:
1517:
1500:
1496:
1488:
1485:
1455:
1447:
1418:
1410:
1393:
1389:
1381:
1364:
1360:
1352:
1337:
1330:
1329:
1328:
1307:
1299:
1295:
1291:
1286:
1282:
1275:
1272:
1267:
1263:
1259:
1256:
1251:
1247:
1243:
1240:
1237:
1232:
1228:
1224:
1221:
1216:
1212:
1208:
1205:
1202:
1200:
1193:
1189:
1181:
1173:
1169:
1165:
1160:
1156:
1149:
1146:
1141:
1137:
1133:
1130:
1125:
1121:
1117:
1114:
1111:
1106:
1102:
1098:
1095:
1090:
1086:
1082:
1079:
1076:
1074:
1067:
1063:
1055:
1047:
1043:
1039:
1034:
1030:
1023:
1020:
1015:
1011:
1007:
1004:
1001:
999:
992:
988:
980:
974:
970:
952:
948:
945:
942:
925:
918:
915:
912:
910:
903:
899:
891:
888:
885:
883:
876:
872:
860:
859:
858:
856:
852:
848:
844:
839:
837:
833:
829:
807:
803:
780:
776:
753:
749:
739:
732:
728:
726:
724:
717:
714:is the local
694:
692:
676:
669:
667:
663:
647:
640:
638:
614:
606:
583:
575:
574:
573:
556:
545:
541:
538:
535:
524:
520:compass
511:
510:
509:
487:
478:
475:
464:
460:
457:
449:
446:
435:
431:
428:
425:
422:
419:
416:
410:
408:
395:
391:
388:
370:
366:
363:
355:
352:
349:
346:
343:
340:
337:
334:
331:
325:
322:
319:
316:
313:
307:
305:
292:
288:
285:
274:
273:
272:
270:
242:
228:
224:
221:
216:
213:
210:
207:
204:
201:
198:
192:
181:
177:
174:
167:
166:
165:
162:
158:
151:
146:
137:
135:
133:
129:
125:
121:
117:
113:
109:
105:
100:
97:
93:
89:
85:
77:
75:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
18:Solar azimuth
2492:Solar energy
2406:
2393:
2381:
2368:
2359:
2345:cite journal
2312:
2309:Solar Energy
2308:
2279:
2256:. Retrieved
2252:the original
2236:
2205:
2037:
1876:
1820:
1602:
1326:
853:rather than
840:
825:
730:
722:
720:
571:
506:
257:
160:
156:
149:
145:inverse sine
141:
108:solar energy
101:
81:
46:defines the
31:
29:
1486:in minutes,
106:, e.g. for
2476:Categories
2258:2013-05-01
2188:References
2157:Solar time
2147:Hour angle
836:Fortran 90
666:solar time
662:hour angle
2337:0038-092X
2121:analemmas
1967:γ
1889:γ
1855:γ
1792:−
1776:−
1744:γ
1526:λ
1497:ϕ
1390:λ
1361:ϕ
1338:δ
1296:λ
1292:−
1283:λ
1276:
1264:ϕ
1260:
1248:ϕ
1244:
1229:ϕ
1225:
1213:ϕ
1209:
1170:λ
1166:−
1157:λ
1150:
1138:ϕ
1134:
1122:ϕ
1118:
1112:−
1103:ϕ
1099:
1087:ϕ
1083:
1044:λ
1040:−
1031:λ
1024:
1012:ϕ
1008:
943:−
916:−
900:λ
889:δ
873:ϕ
702:Φ
677:δ
615:θ
584:ϕ
546:ϕ
542:−
525:ϕ
482:Φ
479:
465:θ
461:
453:Φ
450:
436:θ
432:
426:−
423:δ
420:
396:ϕ
392:
371:θ
367:
359:Φ
356:
350:δ
347:
338:
332:−
329:Φ
326:
320:δ
317:
293:ϕ
289:
229:θ
225:
217:δ
214:
205:
199:−
182:ϕ
178:
159:= arcsin
132:clockwise
68:elevation
2167:Sun path
2131:See also
2125:sun path
830:and the
716:latitude
120:formulas
72:altitude
62:(or its
2466:PVCDROM
2317:Bibcode
2172:Sunrise
1482:is the
660:is the
635:is the
152:= sin y
147:, i.e.
124:Landsat
56:horizon
42:. This
36:azimuth
34:is the
2335:
2287:
2244:
2212:
2182:Zenith
2177:Sunset
1821:where
1327:where
66:solar
832:atan2
112:north
88:Earth
84:south
2444:and
2373:USGS
2351:link
2333:ISSN
2285:ISBN
2242:ISBN
2210:ISBN
2092:and
845:and
795:and
128:USGS
116:NREL
96:west
92:east
30:The
2487:Sun
2325:doi
1273:cos
1257:cos
1241:cos
1222:sin
1206:sin
1147:cos
1131:cos
1115:sin
1096:sin
1080:cos
1021:sin
1005:cos
539:360
476:cos
458:sin
447:sin
429:cos
417:sin
389:cos
364:sin
353:sin
344:cos
335:cos
323:cos
314:sin
286:cos
222:sin
211:cos
202:sin
175:sin
154:or
50:'s
48:Sun
2478::
2371:.
2367:.
2347:}}
2343:{{
2331:.
2323:.
2313:76
2311:.
2299:^
2267:^
2224:^
2196:^
2065:,
975:60
946:12
919:15
768:,
725:.
74:.
2375:.
2353:)
2339:.
2327::
2319::
2293:.
2261:.
2218:.
2105:z
2101:S
2078:y
2074:S
2051:x
2047:S
2022:)
2017:x
2013:S
2009:,
2004:y
2000:S
1996:(
1992:2
1989:n
1986:a
1983:t
1980:a
1976:=
1971:s
1944:)
1939:y
1935:S
1931:,
1926:x
1922:S
1918:(
1914:2
1911:n
1908:a
1905:t
1902:a
1898:=
1893:s
1859:s
1832:Z
1805:)
1800:y
1796:S
1789:,
1784:x
1780:S
1773:(
1769:2
1766:n
1763:a
1760:t
1757:a
1753:=
1748:s
1733:,
1721:)
1716:z
1712:S
1708:(
1704:s
1701:o
1698:c
1695:a
1691:=
1688:Z
1665:1
1662:=
1657:2
1652:z
1648:S
1644:+
1639:2
1634:y
1630:S
1626:+
1621:2
1616:x
1612:S
1585:z
1581:S
1577:,
1572:y
1568:S
1564:,
1559:x
1555:S
1530:o
1501:o
1467:n
1464:i
1461:m
1456:E
1430:T
1427:M
1424:G
1419:T
1394:s
1365:s
1308:.
1305:)
1300:o
1287:s
1279:(
1268:s
1252:o
1238:+
1233:s
1217:o
1203:=
1194:z
1190:S
1182:,
1179:)
1174:o
1161:s
1153:(
1142:s
1126:o
1107:s
1091:o
1077:=
1068:y
1064:S
1056:,
1053:)
1048:o
1035:s
1027:(
1016:s
1002:=
993:x
989:S
981:,
978:)
971:/
964:n
961:i
958:m
953:E
949:+
937:T
934:M
931:G
926:T
922:(
913:=
904:s
892:,
886:=
877:s
808:z
804:S
781:y
777:S
754:x
750:S
648:h
620:s
589:s
557:.
551:s
536:=
530:s
488:.
470:s
441:s
411:=
401:s
376:s
341:h
308:=
298:s
265:h
261:h
243:.
234:s
208:h
193:=
187:s
161:y
157:x
150:x
20:)
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