20:
2256:. As the graph shows, a sundial can be up to about 16 minutes fast or slow, compared with a clock. Since the Earth rotates at a mean speed of one degree every four minutes, relative to the Sun, this 16-minute displacement corresponds to a shift eastward or westward of about four degrees in the apparent position of the Sun, compared with its mean position. A westward shift causes the sundial to be ahead of the clock.
1364:
2230:
3097:
1793:) is also discouraged for having an error of up to 0.28°. An additional error of up to 0.5° can occur in all equations around the equinoxes if not using a decimal place when selecting N to adjust for the time after UT midnight for the beginning of that day. So the above equation can have up to 2.0° of error, about four times the Sun's angular width, depending on how it is used.
1498:, in early July. This makes processes like the variation of the solar declination happen faster in January than in July. On the graph, this makes the minima more acute than the maxima. Also, since perihelion and aphelion do not happen on the exact dates as the solstices, the maxima and minima are slightly asymmetrical. The rates of change before and after are not quite equal.
1789:−1). The number 10, in (N+10), is the approximate number of days after the December solstice to January 1. This equation overestimates the declination near the September equinox by up to +1.5°. The sine function approximation by itself leads to an error of up to 0.26° and has been discouraged for use in solar energy applications. The 1971 Spencer formula (based on a
3121:
3073:
3109:
3085:
2243:
In addition to the annual north–south oscillation of the Sun's apparent position, corresponding to the variation of its declination described above, there is also a smaller but more complex oscillation in the east–west direction. This is caused by the tilt of the Earth's axis, and also by changes in
1673:
is small, its orbit can be approximated as a circle which causes up to 1° of error. The circle approximation means the EL would be 90° ahead of the solstices in Earth's orbit (at the equinoxes), so that sin(EL) can be written as sin(90+NDS)=cos(NDS) where NDS is the number of days after the
December
2216:
of light in the atmosphere, which causes the apparent angle of elevation of the Sun as seen by an observer to be higher than the actual angle of elevation, especially at low Sun elevations. For example, when the Sun is at an elevation of 10°, it appears to be at 10.1°. The Sun's declination can be
2198:
More complicated algorithms correct for changes to the ecliptic longitude by using terms in addition to the 1st-order eccentricity correction above. They also correct the 23.44° obliquity which changes very slightly with time. Corrections may also include the effects of the moon in offsetting the
1525:
by astronomers) is the angle between the Earth's axis and a line perpendicular to the Earth's orbit. The Earth's axial tilt changes slowly over thousands of years but its current value of about ε = 23°44' is nearly constant, so the change in solar declination during one year is nearly the same as
1478:
If the 90° axial tilt is decreased, then the absolute maximum and minimum values of the declination would decrease, to equal the axial tilt. Also, the shapes of the maxima and minima on the graph would become less acute, being curved to resemble the maxima and minima of a sine wave. However, even
2589:
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.
2183:
of the Earth's orbit. The eccentricity varies very slowly over time, but for dates fairly close to the present, it can be considered to be constant. The largest errors in this equation are less than ± 0.2°, but are less than ± 0.03° for a given year if the number 10 is adjusted up or down in
2328:, plotted horizontally. Usually, the scales are chosen so that equal distances on the diagram represent equal angles in both directions on the celestial sphere. Thus 4 minutes (more precisely 3 minutes, 56 seconds), in the equation of time, are represented by the same distance as 1° in the
2184:
fractional days as determined by how far the previous year's
December solstice occurred before or after noon on December 22. These accuracies are compared to NOAA's advanced calculations which are based on the 1999 Jean Meeus algorithm that is accurate to within 0.01°.
2263:, using the word "equation" in a somewhat archaic sense meaning "correction". The oscillation is measured in units of time, minutes and seconds, corresponding to the amount that a sundial would be ahead of a clock. The equation of time can be positive or negative.
2203:
is applied, which depends on the observer's distance away from the center of the Earth. This correction is less than 0.0025°. The error in calculating the position of the center of the Sun can be less than 0.00015°. For comparison, the Sun's width is about 0.5°.
1466:
with time. Eventually, the Sun would be directly above the South Pole, with a declination of −90°; then it would start to move northward at a constant speed. Thus, the graph of solar declination, as seen from this highly tilted Earth, would resemble a
2244:
the speed of its orbital motion around the Sun produced by the elliptical shape of the orbit. The principal effects of this east–west oscillation are variations in the timing of events such as sunrise and sunset, and in the reading of a
2358:
Some analemmas are marked to show the position of the Sun on the graph on various dates, a few days apart, throughout the year. This enables the analemma to be used to make simple analog computations of quantities such as the times and
1367:
The path of the Sun over the celestial sphere through the course of the day for an observer at 56°N latitude. The Sun's path changes with its declination during the year. The intersections of the curves with the horizontal axis show
1060:
601:
1776:
776:
1123:
941:
711:
423:
364:
2166:
2005:
1660:
1267:
1222:
301:
1177:
2175:−1) and can include decimals to adjust for local times later or earlier in the day. The number 2, in (N-2), is the approximate number of days after January 1 to the Earth's
879:
985:
803:
637:
961:
1501:
The graph of apparent solar declination is therefore different in several ways from a sine wave. Calculating it accurately involves some complexity, as shown below.
823:
3040:
2963:
2199:
Earth's position from the center of the pair's orbit around the Sun. After obtaining the declination relative to the center of the Earth, a further correction for
1796:
The declination can be more accurately calculated by not making the two approximations, using the parameters of the Earth's orbit to more accurately estimate EL:
1337:
1313:
1289:
843:
514:
494:
466:
446:
2233:
The equation of time — above the axis a sundial will appear fast relative to a clock showing local mean time, and below the axis a sundial will appear slow.
997:
2792:
2893:
3146:
2618:
2313:, which resembles a figure-8. An analemma can be pictured by superimposing photographs taken at the same time of day, a few days apart for a
533:
165:, for the observer's local time and location. This is the coordinate system normally used to calculate the position of the Sun in terms of
2850:
2822:
1680:
2682:
2506:
732:
2309:
is also occasionally, but rarely, used in other contexts.) It can be considered as an image of the Sun's apparent motion during a
2221:, to calculate its azimuth and also its true elevation, which can then be corrected for refraction to give its apparent position.
1075:
893:
255:, the number of days (positive or negative, including fractional days) since Greenwich noon, Terrestrial Time, on 1 January 2000 (
19:
2997:
1674:
solstice. By also using the approximation that arcsin is close to d·cos(NDS), the following frequently used formula is obtained:
2272:
373:
of the Sun (actually, of the Earth in its orbit around the Sun, but it is convenient to pretend the Sun orbits the Earth), is:
1533:, the angle between the rays of the Sun and the plane of the Earth's equator reaches its maximum value of 23°44'. Therefore, δ
2407:
2412:
657:
379:
320:
3063:
1479:
when the axial tilt equals that of the actual Earth, the maxima and minima remain more acute than those of a sine wave.
1353:
854:
162:
155:
79:
964:
143:
To find the Sun's position for a given location at a given time, one may therefore proceed in three steps as follows:
2016:
30:
1802:
1565:
1228:
1183:
3007:
850:
644:
607:
524:
148:
3011:
269:
3050:
2715:
2698:
991:
To get RA at the right quadrant on computer programs use double argument Arctan function such as ATAN2(y,x)
2641:
236:, to a precision of about 0°.01 (36″), for dates between 1950 and 2050. Similar equations are coded into a
2213:
469:
2738:
2422:
2892:
Blanco-Muriel, Manuel; Alarcón-Padilla, Diego C; López-Moratalla, Teodoro; Lara-Coira, Martín (2001).
1147:
2908:
2765:
Sproul, Alistair B. (2007). "Derivation of the solar geometric relationships using vector analysis".
2553:
2471:
2180:
1670:
647:
of the Sun never exceeds 0.00033° (a little over 1″), and the distance of the Sun from the Earth, in
224:
133:
2796:
1423:
of 23.44°, but one lobe of the wave is several days longer than the other, among other differences.
3125:
2463:
2417:
2397:
2384:
2383:
We see light from the Sun about 20 angle seconds from where the Sun is when the light is seen. See
2333:
311:
245:
170:
113:
3113:
3101:
3046:
2730:
2569:
2543:
1666:
1459:
517:
241:
166:
864:
2678:
2614:
2531:
2502:
1549:
1401:
1381:
1138:
970:
788:
648:
616:
256:
233:
2846:
2672:
1517:, is the angle between the rays of the Sun and the plane of the Earth's equator. The Earth's
946:
3077:
2959:
2916:
2774:
2608:
2561:
2442:
2427:
2325:
2302:
2284:
2238:
2192:
2188:
1483:
808:
103:
87:
2751:
2645:
2253:
2218:
1377:
1134:
884:
2171:
N is the number of days since midnight UT as
January 1 begins (i.e. the days part of the
3027:
2912:
2557:
2607:
U.S. Naval
Observatory; U.K. Hydrographic Office, H.M. Nautical Almanac Office (2008).
1790:
1782:
1487:
1451:
1431:
1340:
1322:
1298:
1274:
828:
499:
479:
451:
431:
307:
117:
2920:
2339:
An analemma is drawn as it would be seen in the sky by an observer looking upward. If
3135:
3056:
2573:
2565:
2310:
2187:(The above formula is related to a reasonably simple and accurate calculation of the
1468:
1435:
1397:
1385:
1316:
24:
2671:
U.S. Naval
Observatory Nautical Almanac Office (1992). P. Kenneth Seidelmann (ed.).
3141:
3089:
2467:
2305:, relative to its mean position, as seen from a fixed location on Earth. (The word
2172:
1786:
1434:
around the Sun, and if its axis were tilted 90°, so that the axis itself is on the
370:
202:
3053:
website. Positions of Solar System objects based on the INPOP series ephemerides.
2778:
2634:
2633:
Much the same set of equations, covering the years 1800 to 2200, can be found at
2871:
2329:
2280:
1475:
between plus and minus 90°, with linear segments between the maxima and minima.
1404:, when its value is the negative of the axial tilt. This variation produces the
1389:
1363:
1066:
260:
198:
194:
129:
99:
2595:
2229:
2212:
The declination calculations described above do not include the effects of the
2935:
2494:
2432:
2176:
1518:
1491:
1455:
1447:
1412:
1393:
858:
723:
186:
3017:
45:
32:
2818:
2649:
2371:. Analemmas without date markings are used to correct the time indicated by
1420:
1416:
1055:{\displaystyle \alpha =\arctan 2(\cos \epsilon \sin \lambda ,\cos \lambda )}
190:
182:
137:
2351:. This is usually done even when the analemma is marked on a geographical
2301:
is a diagram that shows the annual variation of the Sun's position on the
2259:
Since the main effect of this oscillation concerns time, it is called the
2437:
2402:
2298:
2292:
2276:
2200:
1530:
1495:
1427:
174:
125:
121:
107:
1400:, then decreases until reaching its minimum (−23.44° or -23°26') on the
2983:
2372:
2364:
2360:
2245:
1545:
1369:
1292:
237:
206:
3039:
website. Very accurate positions of Solar System objects based on the
2368:
1472:
1443:
1439:
1405:
596:{\displaystyle \lambda =L+1.915^{\circ }\sin g+0.020^{\circ }\sin 2g}
3036:
3032:
2271:
1669:(essentially, the Earth's position in its orbit). Since the Earth's
3084:
16:
Calculating the Sun's location in the sky at a given time and place
2648:. Graphs of the error of these equations, compared to an accurate
2548:
2355:, on which the continents, etc., are shown with west to the left.
2352:
2340:
2270:
2249:
2228:
1362:
83:
18:
1462:
at a constant rate, so that the solar declination would decrease
1450:, so its declination would be +90°. For the next few months, the
3021:
2638:
2344:
2314:
1771:{\displaystyle \delta _{\odot }=-23.44^{\circ }\cdot \cos \left}
95:
75:
3014:. An interactive calculator showing the Sun's path in the sky.
2321:
1781:
where N is the day of the year beginning with N=0 at midnight
1510:
846:
229:
91:
71:
3001:
771:{\displaystyle \epsilon =23.439^{\circ }-0.0000004^{\circ }n}
2336:
at a mean speed of 1° every 4 minutes, relative to the Sun.
1559:
The Sun's declination at any given moment is calculated by:
1426:
The following phenomena would occur if Earth were a perfect
1118:{\displaystyle \delta =\arcsin(\sin \epsilon \sin \lambda )}
936:{\displaystyle \alpha =\arctan(\cos \epsilon \tan \lambda )}
2936:"Solar Position Algorithm for Solar Radiation Applications"
228:, can be used to calculate the apparent coordinates of the
2731:"Fourier series representation of the position of the sun"
2677:. University Science Books, Mill Valley, CA. p. 12.
1376:
The Sun appears to move northward during the northern
706:{\displaystyle R=1.00014-0.01671\cos g-0.00014\cos 2g}
418:{\displaystyle g=357.528^{\circ }+0.9856003^{\circ }n}
359:{\displaystyle L=280.460^{\circ }+0.9856474^{\circ }n}
120:, so that the Sun appears to move across the sky in a
3061:
2497:(1991). "Chapter 12: Transformation of Coordinates".
2320:
An analemma can also be considered as a graph of the
2019:
1805:
1683:
1568:
1325:
1301:
1277:
1231:
1186:
1150:
1078:
1000:
973:
949:
896:
867:
831:
811:
791:
735:
660:
619:
536:
502:
482:
454:
434:
382:
323:
272:
2010:
which can be simplified by evaluating constants to:
1785:(UT) as January 1 begins (i.e. the days part of the
1548:, the center of the Sun appears to pass through the
3020:, at the NOAA Earth System Research Laboratories's
1415:of the Sun's declination during a year resembles a
1372:
in degrees from North where the Sun rises and sets.
726:is not obtained elsewhere, it can be approximated:
173:, and the two parameters can be used to depict the
2674:Explanatory Supplement to the Astronomical Almanac
2160:
1999:
1770:
1654:
1331:
1307:
1283:
1261:
1216:
1171:
1117:
1054:
979:
955:
935:
873:
837:
817:
797:
770:
705:
631:
595:
508:
488:
460:
440:
417:
358:
295:
1392:reaches a maximum equal to the angle of Earth's
468:in the range 0° to 360° by adding or subtracting
240:90 routine in Ref. and are used to calculate the
2179:. The number 0.0167 is the current value of the
2964:National Oceanic and Atmospheric Administration
2466:, the Earth's orbit would be stated as being a
2161:{\displaystyle \delta _{\odot }=-\arcsin \left}
1537:= +23°44' at the northern summer solstice and δ
1490:. Earth moves more rapidly around the Sun near
3047:General ephemerides of the solar system bodies
3028:NOAA's declination and sun position calculator
2000:{\displaystyle \delta _{\odot }=\arcsin \left}
1655:{\displaystyle \delta _{\odot }=\arcsin \left}
1262:{\displaystyle Z=R\sin \epsilon \sin \lambda }
1217:{\displaystyle Y=R\cos \epsilon \sin \lambda }
98:, the Sun appears to move with respect to the
2716:"Selected Astronomical Constants, 2015 (TXT)"
2699:"Selected Astronomical Constants, 2015 (PDF)"
1442:). At one date in the year, the Sun would be
8:
3000:, at National Renewable Energy Laboratory's
2718:. US Naval Observatory. 2014. p. K6–K7.
2704:. US Naval Observatory. 2014. p. K6–K7.
2596:https://doi.org/10.1016/j.renene.2021.03.047
2841:
2839:
2585:
2583:
1541:= −23°44' at the southern summer solstice.
248:as observed from the surface of the Earth.
2613:. U.S. Govt. Printing Office. p. C5.
2610:The Astronomical Almanac for the Year 2010
2324:, usually plotted vertically, against the
1359:Declination of the Sun as seen from Earth
124:that depends on the observer's geographic
2547:
2534:(2013). "The Sun's position in the sky".
2526:
2524:
2522:
2520:
2518:
2118:
2097:
2065:
2024:
2018:
1952:
1946:
1918:
1912:
1879:
1873:
1848:
1810:
1804:
1730:
1724:
1704:
1688:
1682:
1611:
1573:
1567:
1324:
1300:
1276:
1230:
1185:
1149:
1077:
999:
972:
948:
895:
866:
830:
810:
790:
759:
746:
734:
659:
618:
575:
553:
535:
501:
481:
453:
433:
406:
393:
381:
347:
334:
322:
296:{\displaystyle n=\mathrm {JD} -2451545.0}
279:
271:
3068:
2486:
2455:
2960:"Atmospheric Refraction Approximation"
2747:
2736:
1137:rectangular equatorial coordinates in
7:
2501:. Richmond, VA: Willmann Bell, Inc.
263:for the desired time is known, then
147:calculate the Sun's position in the
3057:Solar position in R. Insol package.
2934:Ibrahim Reda & Afshin Andreas.
106:, along a circular path called the
2851:Earth System Research Laboratories
2823:Earth System Research Laboratories
1130:Rectangular equatorial coordinates
283:
280:
62:a.m. local time on a March morning
14:
234:mean equinox and ecliptic of date
3119:
3107:
3095:
3083:
3071:
1458:at constant speed, crossing the
1291:axis is in the direction of the
1172:{\displaystyle X=R\cos \lambda }
845:form a complete position of the
853:. This can be converted to the
3002:Renewable Resource Data Center
2639:U.S. Naval Observatory website
2408:Effect of Sun angle on climate
1494:, in early January, than near
1112:
1091:
1049:
1016:
930:
909:
310:of the Sun, corrected for the
181:This calculation is useful in
1:
2921:10.1016/s0038-092x(00)00156-0
2635:Approximate Solar Coordinates
3147:Dynamics of the Solar System
2894:"Computing the Solar Vector"
2779:10.1016/j.renene.2006.05.001
1354:Horizontal coordinate system
855:equatorial coordinate system
516:are really to be evaluated (
163:horizontal coordinate system
156:equatorial coordinate system
2847:"Solar Calculation Details"
2536:European Journal of Physics
2413:Newcomb's Tables of the Sun
3163:
3022:Global Monitoring Division
2566:10.1088/0143-0807/34/3/633
2343:is shown at the top, then
2290:
2236:
1396:(23.44° or 23°26') on the
1351:
851:ecliptic coordinate system
222:These equations, from the
149:ecliptic coordinate system
136:depends on the geographic
74:is a function of both the
2872:"Astronomical Algorithms"
1523:obliquity of the ecliptic
1471:rather than a sine wave,
874:{\displaystyle \epsilon }
859:obliquity of the ecliptic
724:obliquity of the ecliptic
718:Obliquity of the ecliptic
476:— which is to say,
2998:Solar Position Algorithm
980:{\displaystyle \lambda }
798:{\displaystyle \lambda }
632:{\displaystyle \beta =0}
128:. The time when the Sun
3008:Sun Position Calculator
2819:"NOAA Solar Calculator"
2594:, 172, 1333-1340. DOI:
2499:Astronomical Algorithms
2398:Solar annual aberration
2385:Solar annual aberration
1509:The declination of the
956:{\displaystyle \alpha }
2746:Cite journal requires
2729:J. W. Spencer (1971).
2288:
2234:
2208:Atmospheric refraction
2162:
2001:
1772:
1656:
1544:At the moment of each
1526:during the next year.
1454:would move toward the
1373:
1348:Horizontal coordinates
1333:
1309:
1285:
1263:
1218:
1173:
1119:
1056:
981:
957:
937:
875:
839:
819:
818:{\displaystyle \beta }
799:
782:Equatorial coordinates
772:
707:
633:
610:of the Sun is nearly:
597:
510:
490:
462:
442:
419:
360:
297:
116:about its axis causes
63:
3018:NOAA Solar Calculator
2652:, can also be viewed.
2423:Solar elevation angle
2274:
2232:
2217:used, along with its
2191:, which is described
2163:
2002:
1773:
1657:
1366:
1352:Further information:
1334:
1310:
1286:
1264:
1219:
1174:
1120:
1057:
982:
958:
938:
876:
840:
820:
800:
773:
708:
634:
598:
511:
491:
463:
443:
420:
361:
298:
251:Start by calculating
94:over the course of a
22:
2661:Meeus (1991), p. 152
2472:orbital eccentricity
2017:
1803:
1681:
1671:orbital eccentricity
1566:
1341:North celestial pole
1323:
1299:
1275:
1229:
1184:
1148:
1076:
998:
971:
947:
894:
865:
829:
809:
789:
733:
658:
617:
534:
500:
480:
452:
432:
380:
321:
270:
225:Astronomical Almanac
218:Ecliptic coordinates
213:Approximate position
2913:2001SoEn...70..431B
2558:2013EJPh...34..633J
2464:celestial mechanics
2418:Solar azimuth angle
1460:circles of latitude
881:, and continuing:
857:by calculating the
312:aberration of light
246:solar azimuth angle
171:solar azimuth angle
80:geographic location
68:position of the Sun
42: /
2984:Sundial#Noon marks
2644:2016-01-31 at the
2532:Jenkins, Alejandro
2379:Light-time effects
2289:
2287:to the same scale
2235:
2158:
1997:
1768:
1667:ecliptic longitude
1652:
1374:
1329:
1305:
1281:
1259:
1214:
1169:
1139:astronomical units
1115:
1052:
977:
953:
933:
871:
835:
815:
795:
768:
703:
649:astronomical units
629:
593:
525:ecliptic longitude
506:
486:
458:
438:
415:
356:
293:
242:solar zenith angle
167:solar zenith angle
82:of observation on
64:
2620:978-0-7077-4082-9
2322:Sun's declination
1961:
1927:
1888:
1739:
1550:celestial equator
1444:directly overhead
1402:December solstice
1382:celestial equator
1339:axis towards the
1332:{\displaystyle Z}
1308:{\displaystyle Y}
1284:{\displaystyle X}
838:{\displaystyle R}
645:ecliptic latitude
608:ecliptic latitude
509:{\displaystyle g}
489:{\displaystyle L}
472:of 360° as needed
461:{\displaystyle g}
441:{\displaystyle L}
3154:
3124:
3123:
3122:
3112:
3111:
3110:
3100:
3099:
3098:
3088:
3087:
3076:
3075:
3074:
3067:
2986:
2981:
2975:
2974:
2972:
2970:
2956:
2950:
2949:
2947:
2945:
2940:
2931:
2925:
2924:
2898:
2889:
2883:
2882:
2880:
2878:
2868:
2862:
2861:
2859:
2857:
2843:
2834:
2833:
2831:
2829:
2815:
2809:
2808:
2806:
2804:
2795:. Archived from
2789:
2783:
2782:
2773:(7): 1187–1205.
2767:Renewable Energy
2762:
2756:
2755:
2749:
2744:
2742:
2734:
2726:
2720:
2719:
2712:
2706:
2705:
2703:
2695:
2689:
2688:
2668:
2662:
2659:
2653:
2631:
2625:
2624:
2604:
2598:
2592:Renewable Energy
2587:
2578:
2577:
2551:
2528:
2513:
2512:
2491:
2475:
2460:
2443:Sunrise equation
2428:Solar irradiance
2326:equation of time
2303:celestial sphere
2285:equation of time
2261:equation of time
2248:compared with a
2239:Equation of time
2225:Equation of time
2189:Equation of Time
2167:
2165:
2164:
2159:
2157:
2153:
2152:
2148:
2147:
2143:
2142:
2138:
2123:
2122:
2102:
2101:
2089:
2085:
2070:
2069:
2029:
2028:
2006:
2004:
2003:
1998:
1996:
1992:
1991:
1987:
1986:
1982:
1981:
1977:
1962:
1957:
1956:
1947:
1928:
1923:
1922:
1913:
1908:
1904:
1889:
1884:
1883:
1874:
1858:
1854:
1853:
1852:
1815:
1814:
1777:
1775:
1774:
1769:
1767:
1763:
1762:
1758:
1740:
1735:
1734:
1725:
1709:
1708:
1693:
1692:
1665:where EL is the
1661:
1659:
1658:
1653:
1651:
1647:
1646:
1642:
1621:
1617:
1616:
1615:
1578:
1577:
1338:
1336:
1335:
1330:
1314:
1312:
1311:
1306:
1290:
1288:
1287:
1282:
1268:
1266:
1265:
1260:
1223:
1221:
1220:
1215:
1178:
1176:
1175:
1170:
1124:
1122:
1121:
1116:
1061:
1059:
1058:
1053:
986:
984:
983:
978:
962:
960:
959:
954:
942:
940:
939:
934:
880:
878:
877:
872:
844:
842:
841:
836:
824:
822:
821:
816:
804:
802:
801:
796:
777:
775:
774:
769:
764:
763:
751:
750:
712:
710:
709:
704:
638:
636:
635:
630:
602:
600:
599:
594:
580:
579:
558:
557:
515:
513:
512:
507:
495:
493:
492:
487:
475:
467:
465:
464:
459:
447:
445:
444:
439:
424:
422:
421:
416:
411:
410:
398:
397:
365:
363:
362:
357:
352:
351:
339:
338:
302:
300:
299:
294:
286:
114:Earth's rotation
104:celestial sphere
61:
57:
56:
54:
53:
52:
47:
46:8.283°N 98.600°E
43:
40:
39:
38:
35:
3162:
3161:
3157:
3156:
3155:
3153:
3152:
3151:
3132:
3131:
3130:
3120:
3118:
3108:
3106:
3096:
3094:
3082:
3072:
3070:
3062:
3033:HORIZONS System
3012:pveducation.org
2994:
2989:
2982:
2978:
2968:
2966:
2958:
2957:
2953:
2943:
2941:
2938:
2933:
2932:
2928:
2896:
2891:
2890:
2886:
2876:
2874:
2870:
2869:
2865:
2855:
2853:
2845:
2844:
2837:
2827:
2825:
2817:
2816:
2812:
2802:
2800:
2799:on 9 March 2012
2791:
2790:
2786:
2764:
2763:
2759:
2745:
2735:
2728:
2727:
2723:
2714:
2713:
2709:
2701:
2697:
2696:
2692:
2685:
2670:
2669:
2665:
2660:
2656:
2646:Wayback Machine
2632:
2628:
2621:
2606:
2605:
2601:
2588:
2581:
2530:
2529:
2516:
2509:
2493:
2492:
2488:
2484:
2479:
2478:
2474:of less than 1.
2461:
2457:
2452:
2447:
2393:
2381:
2295:
2269:
2254:local mean time
2241:
2227:
2219:right ascension
2210:
2128:
2124:
2114:
2113:
2109:
2093:
2075:
2071:
2061:
2060:
2056:
2046:
2042:
2020:
2015:
2014:
1967:
1963:
1948:
1945:
1941:
1914:
1894:
1890:
1875:
1872:
1868:
1844:
1840:
1836:
1829:
1825:
1806:
1801:
1800:
1748:
1744:
1726:
1723:
1719:
1700:
1684:
1679:
1678:
1635:
1631:
1607:
1603:
1599:
1592:
1588:
1569:
1564:
1563:
1555:
1540:
1536:
1516:
1507:
1380:, crossing the
1361:
1356:
1350:
1321:
1320:
1297:
1296:
1273:
1272:
1227:
1226:
1182:
1181:
1146:
1145:
1132:
1074:
1073:
996:
995:
969:
968:
963:is in the same
945:
944:
892:
891:
885:Right ascension
863:
862:
827:
826:
807:
806:
787:
786:
784:
755:
742:
731:
730:
720:
656:
655:
615:
614:
571:
549:
532:
531:
527:of the Sun is:
498:
497:
478:
477:
473:
450:
449:
430:
429:
402:
389:
378:
377:
343:
330:
319:
318:
268:
267:
220:
215:
161:convert to the
154:convert to the
132:the observer's
86:'s surface. As
59:
50:
48:
44:
41:
36:
33:
31:
29:
28:
17:
12:
11:
5:
3160:
3158:
3150:
3149:
3144:
3134:
3133:
3129:
3128:
3116:
3104:
3092:
3080:
3060:
3059:
3054:
3044:
3030:
3025:
3015:
3005:
2993:
2992:External links
2990:
2988:
2987:
2976:
2951:
2926:
2907:(5): 431–441.
2884:
2863:
2835:
2810:
2784:
2757:
2748:|journal=
2721:
2707:
2690:
2683:
2663:
2654:
2626:
2619:
2599:
2579:
2542:(3): 633–652.
2514:
2507:
2485:
2483:
2480:
2477:
2476:
2454:
2453:
2451:
2448:
2446:
2445:
2440:
2435:
2430:
2425:
2420:
2415:
2410:
2405:
2400:
2394:
2392:
2389:
2380:
2377:
2291:Main article:
2268:
2265:
2237:Main article:
2226:
2223:
2209:
2206:
2169:
2168:
2156:
2151:
2146:
2141:
2137:
2134:
2131:
2127:
2121:
2117:
2112:
2108:
2105:
2100:
2096:
2092:
2088:
2084:
2081:
2078:
2074:
2068:
2064:
2059:
2055:
2052:
2049:
2045:
2041:
2038:
2035:
2032:
2027:
2023:
2008:
2007:
1995:
1990:
1985:
1980:
1976:
1973:
1970:
1966:
1960:
1955:
1951:
1944:
1940:
1937:
1934:
1931:
1926:
1921:
1917:
1911:
1907:
1903:
1900:
1897:
1893:
1887:
1882:
1878:
1871:
1867:
1864:
1861:
1857:
1851:
1847:
1843:
1839:
1835:
1832:
1828:
1824:
1821:
1818:
1813:
1809:
1791:Fourier series
1783:Universal Time
1779:
1778:
1766:
1761:
1757:
1754:
1751:
1747:
1743:
1738:
1733:
1729:
1722:
1718:
1715:
1712:
1707:
1703:
1699:
1696:
1691:
1687:
1663:
1662:
1650:
1645:
1641:
1638:
1634:
1630:
1627:
1624:
1620:
1614:
1610:
1606:
1602:
1598:
1595:
1591:
1587:
1584:
1581:
1576:
1572:
1553:
1538:
1534:
1514:
1506:
1503:
1452:subsolar point
1432:circular orbit
1360:
1357:
1349:
1346:
1345:
1344:
1328:
1304:
1280:
1269:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1224:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1179:
1168:
1165:
1162:
1159:
1156:
1153:
1131:
1128:
1127:
1126:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1063:
1062:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
989:
988:
976:
952:
932:
929:
926:
923:
920:
917:
914:
911:
908:
905:
902:
899:
870:
834:
814:
794:
783:
780:
779:
778:
767:
762:
758:
754:
749:
745:
741:
738:
719:
716:
715:
714:
702:
699:
696:
693:
690:
687:
684:
681:
678:
675:
672:
669:
666:
663:
641:
640:
628:
625:
622:
604:
603:
592:
589:
586:
583:
578:
574:
570:
567:
564:
561:
556:
552:
548:
545:
542:
539:
505:
485:
457:
437:
426:
425:
414:
409:
405:
401:
396:
392:
388:
385:
367:
366:
355:
350:
346:
342:
337:
333:
329:
326:
308:mean longitude
304:
303:
292:
289:
285:
282:
278:
275:
219:
216:
214:
211:
179:
178:
159:
152:
118:diurnal motion
15:
13:
10:
9:
6:
4:
3:
2:
3159:
3148:
3145:
3143:
3140:
3139:
3137:
3127:
3117:
3115:
3105:
3103:
3093:
3091:
3086:
3081:
3079:
3069:
3065:
3058:
3055:
3052:
3048:
3045:
3042:
3041:JPL DE series
3038:
3034:
3031:
3029:
3026:
3023:
3019:
3016:
3013:
3009:
3006:
3003:
2999:
2996:
2995:
2991:
2985:
2980:
2977:
2965:
2961:
2955:
2952:
2937:
2930:
2927:
2922:
2918:
2914:
2910:
2906:
2902:
2895:
2888:
2885:
2873:
2867:
2864:
2852:
2848:
2842:
2840:
2836:
2824:
2820:
2814:
2811:
2798:
2794:
2788:
2785:
2780:
2776:
2772:
2768:
2761:
2758:
2753:
2740:
2732:
2725:
2722:
2717:
2711:
2708:
2700:
2694:
2691:
2686:
2684:0-935702-68-7
2680:
2676:
2675:
2667:
2664:
2658:
2655:
2651:
2647:
2643:
2640:
2636:
2630:
2627:
2622:
2616:
2612:
2611:
2603:
2600:
2597:
2593:
2586:
2584:
2580:
2575:
2571:
2567:
2563:
2559:
2555:
2550:
2545:
2541:
2537:
2533:
2527:
2525:
2523:
2521:
2519:
2515:
2510:
2508:0-943396-35-2
2504:
2500:
2496:
2490:
2487:
2481:
2473:
2469:
2465:
2459:
2456:
2449:
2444:
2441:
2439:
2436:
2434:
2431:
2429:
2426:
2424:
2421:
2419:
2416:
2414:
2411:
2409:
2406:
2404:
2401:
2399:
2396:
2395:
2390:
2388:
2386:
2378:
2376:
2374:
2370:
2366:
2362:
2356:
2354:
2350:
2346:
2342:
2337:
2335:
2334:Earth rotates
2331:
2327:
2323:
2318:
2316:
2312:
2308:
2304:
2300:
2294:
2286:
2282:
2278:
2273:
2266:
2264:
2262:
2257:
2255:
2251:
2247:
2240:
2231:
2224:
2222:
2220:
2215:
2207:
2205:
2202:
2196:
2194:
2190:
2185:
2182:
2178:
2174:
2154:
2149:
2144:
2139:
2135:
2132:
2129:
2125:
2119:
2115:
2110:
2106:
2103:
2098:
2094:
2090:
2086:
2082:
2079:
2076:
2072:
2066:
2062:
2057:
2053:
2050:
2047:
2043:
2039:
2036:
2033:
2030:
2025:
2021:
2013:
2012:
2011:
1993:
1988:
1983:
1978:
1974:
1971:
1968:
1964:
1958:
1953:
1949:
1942:
1938:
1935:
1932:
1929:
1924:
1919:
1915:
1909:
1905:
1901:
1898:
1895:
1891:
1885:
1880:
1876:
1869:
1865:
1862:
1859:
1855:
1849:
1845:
1841:
1837:
1833:
1830:
1826:
1822:
1819:
1816:
1811:
1807:
1799:
1798:
1797:
1794:
1792:
1788:
1784:
1764:
1759:
1755:
1752:
1749:
1745:
1741:
1736:
1731:
1727:
1720:
1716:
1713:
1710:
1705:
1701:
1697:
1694:
1689:
1685:
1677:
1676:
1675:
1672:
1668:
1648:
1643:
1639:
1636:
1632:
1628:
1625:
1622:
1618:
1612:
1608:
1604:
1600:
1596:
1593:
1589:
1585:
1582:
1579:
1574:
1570:
1562:
1561:
1560:
1557:
1551:
1547:
1542:
1532:
1527:
1524:
1520:
1512:
1504:
1502:
1499:
1497:
1493:
1489:
1485:
1484:Earth's orbit
1480:
1476:
1474:
1470:
1469:triangle wave
1465:
1461:
1457:
1453:
1449:
1445:
1441:
1437:
1436:orbital plane
1433:
1429:
1424:
1422:
1418:
1414:
1409:
1407:
1403:
1399:
1398:June solstice
1395:
1391:
1387:
1386:March equinox
1383:
1379:
1371:
1365:
1358:
1355:
1347:
1342:
1326:
1318:
1317:June Solstice
1315:axis towards
1302:
1294:
1293:March equinox
1278:
1270:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1225:
1211:
1208:
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1180:
1166:
1163:
1160:
1157:
1154:
1151:
1144:
1143:
1142:
1140:
1136:
1129:
1109:
1106:
1103:
1100:
1097:
1094:
1088:
1085:
1082:
1079:
1072:
1071:
1070:
1068:
1046:
1043:
1040:
1037:
1034:
1031:
1028:
1025:
1022:
1019:
1013:
1010:
1007:
1004:
1001:
994:
993:
992:
974:
966:
950:
927:
924:
921:
918:
915:
912:
906:
903:
900:
897:
890:
889:
888:
886:
882:
868:
860:
856:
852:
848:
832:
812:
792:
781:
765:
760:
756:
752:
747:
743:
739:
736:
729:
728:
727:
725:
717:
700:
697:
694:
691:
688:
685:
682:
679:
676:
673:
670:
667:
664:
661:
654:
653:
652:
650:
646:
626:
623:
620:
613:
612:
611:
609:
590:
587:
584:
581:
576:
572:
568:
565:
562:
559:
554:
550:
546:
543:
540:
537:
530:
529:
528:
526:
523:Finally, the
521:
519:
503:
483:
471:
455:
435:
412:
407:
403:
399:
394:
390:
386:
383:
376:
375:
374:
372:
353:
348:
344:
340:
335:
331:
327:
324:
317:
316:
315:
313:
309:
290:
287:
276:
273:
266:
265:
264:
262:
258:
254:
249:
247:
243:
239:
235:
231:
227:
226:
217:
212:
210:
208:
204:
200:
196:
192:
188:
184:
176:
172:
168:
164:
160:
157:
153:
150:
146:
145:
144:
141:
139:
135:
131:
127:
123:
119:
115:
111:
109:
105:
101:
97:
93:
89:
85:
81:
77:
73:
69:
55:
51:8.283; 98.600
27:in Thailand (
26:
25:Phang Nga Bay
23:The Sun over
21:
3126:Solar System
3043:ephemerides.
2979:
2967:. Retrieved
2954:
2942:. Retrieved
2929:
2904:
2901:Solar Energy
2900:
2887:
2875:. Retrieved
2866:
2854:. Retrieved
2826:. Retrieved
2813:
2801:. Retrieved
2797:the original
2787:
2770:
2766:
2760:
2739:cite journal
2724:
2710:
2693:
2673:
2666:
2657:
2629:
2609:
2602:
2591:
2539:
2535:
2498:
2489:
2468:Kepler orbit
2458:
2382:
2357:
2348:
2338:
2319:
2306:
2296:
2260:
2258:
2242:
2211:
2197:
2186:
2181:eccentricity
2173:ordinal date
2170:
2009:
1795:
1787:ordinal date
1780:
1664:
1558:
1543:
1528:
1522:
1521:(called the
1508:
1505:Calculations
1500:
1482:In reality,
1481:
1477:
1463:
1438:(similar to
1425:
1410:
1375:
1135:Right-handed
1133:
1064:
990:
883:
785:
721:
642:
605:
522:
427:
371:mean anomaly
368:
305:
252:
250:
223:
221:
203:solar energy
180:
142:
112:
88:Earth orbits
67:
65:
3114:Outer space
3102:Spaceflight
2969:28 February
2944:28 February
2877:28 February
2856:28 February
2828:28 February
2803:28 February
2495:Meeus, Jean
2330:declination
2281:declination
2279:with solar
1390:declination
1067:declination
261:Julian date
199:climatology
195:meteorology
100:fixed stars
49: /
3136:Categories
2793:"SunAlign"
2482:References
2433:Solar time
2347:is to the
2214:refraction
2177:perihelion
1519:axial tilt
1492:perihelion
1488:elliptical
1473:zigzagging
1456:South Pole
1448:North Pole
1413:line graph
1394:axial tilt
1319:, and the
722:Where the
259:). If the
187:navigation
58:), at 7:00
3078:Astronomy
3049:, at the
3035:, at the
2650:ephemeris
2637:, at the
2574:119282288
2549:1208.1043
2133:−
2120:∘
2107:
2099:∘
2067:∘
2054:
2040:
2034:−
2026:⊙
2022:δ
1972:−
1954:∘
1939:
1930:⋅
1925:π
1920:∘
1881:∘
1866:
1860:⋅
1850:∘
1842:−
1834:
1823:
1812:⊙
1808:δ
1742:⋅
1732:∘
1717:
1711:⋅
1706:∘
1698:−
1690:⊙
1686:δ
1629:
1623:⋅
1613:∘
1605:−
1597:
1586:
1575:⊙
1571:δ
1531:solstices
1421:amplitude
1417:sine wave
1257:λ
1254:
1248:ϵ
1245:
1212:λ
1209:
1203:ϵ
1200:
1167:λ
1164:
1110:λ
1107:
1101:ϵ
1098:
1089:
1080:δ
1047:λ
1044:
1035:λ
1032:
1026:ϵ
1023:
1011:
1002:α
975:λ
951:α
928:λ
925:
919:ϵ
916:
907:
898:α
869:ϵ
813:β
793:λ
761:∘
757:0.0000004
753:−
748:∘
737:ϵ
695:
686:−
680:
671:−
621:β
585:
577:∘
563:
555:∘
538:λ
470:multiples
408:∘
404:0.9856003
395:∘
349:∘
345:0.9856474
336:∘
291:2451545.0
288:−
191:surveying
183:astronomy
138:longitude
3024:website.
3004:website.
2642:Archived
2470:with an
2438:Sun path
2403:Ecliptic
2391:See also
2373:sundials
2361:azimuths
2332:, since
2307:analemma
2299:analemma
2293:Analemma
2277:analemma
2267:Analemma
2252:showing
2201:parallax
1496:aphelion
1464:linearly
1419:with an
1370:azimuths
965:quadrant
943:, where
209:design.
175:Sun path
134:meridian
130:transits
126:latitude
122:Sun path
108:ecliptic
78:and the
3064:Portals
2909:Bibcode
2554:Bibcode
2365:sunrise
2246:sundial
2116:0.98565
2063:0.98565
2048:0.39779
1556:is 0°.
1552:, and δ
1546:equinox
1529:At the
1446:at the
1430:, in a
1406:seasons
1384:on the
849:in the
689:0.00014
674:0.01671
668:1.00014
643:as the
474:
391:357.528
332:280.460
257:J2000.0
238:Fortran
207:sundial
102:on the
70:in the
37:98°36′E
2681:
2617:
2572:
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2369:sunset
2037:arcsin
1959:365.24
1933:0.0167
1886:365.24
1820:arcsin
1583:arcsin
1440:Uranus
1428:sphere
1388:. Its
1378:spring
1295:, the
1271:Where
1086:arcsin
1008:arctan
904:arctan
744:23.439
651:, is:
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314:, is:
205:, and
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34:8°17′N
3090:Stars
3051:IMCCE
3010:, at
2939:(PDF)
2897:(PDF)
2702:(PDF)
2570:S2CID
2544:arXiv
2450:Notes
2353:globe
2349:right
2341:north
2250:clock
2095:1.914
1846:23.44
1702:23.44
1609:23.44
1141:are:
573:0.020
551:1.915
158:, and
84:Earth
2971:2012
2946:2012
2879:2012
2858:2012
2830:2012
2805:2012
2752:help
2679:ISBN
2615:ISBN
2503:ISBN
2367:and
2345:west
2315:year
2311:year
2283:and
2193:here
1065:and
825:and
606:The
496:and
448:and
428:Put
369:The
306:The
244:and
169:and
96:year
90:the
76:time
66:The
3142:Sun
3037:JPL
2917:doi
2775:doi
2562:doi
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2297:An
2275:An
2195:.)
2104:sin
2051:cos
1950:360
1936:sin
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1877:360
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1831:sin
1737:365
1728:360
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1626:sin
1594:sin
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1511:Sun
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1206:sin
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1161:cos
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1029:sin
1020:cos
967:as
922:tan
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847:Sun
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677:cos
582:sin
560:sin
518:mod
230:Sun
92:Sun
72:sky
3138::
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