1. Analemma picture

2. Analemma in Athens (lat=38, lon=23) 10:00 UTC+2, 12:28 UTC + 2 and 15:00 UTC + 2

3. Analemma on the Globe

4. Close-up

5. Motion of the Sun The daily motion of the Sun is westward The yearly motion of the Sun is eastward with respect to the stars

6. The true Sun The true Sun is the usual sun True noon is when the Sun crosses the meridian True time is time measured by the true Sun, i.e., sundial time The true day is the time from one true noon to the next

7. The mean Sun The mean Sun is a virtual body that moves along the celestial equator at constant speed and meets the true Sun at the vernal equinox The mean Sun crosses the meridian at mean noon, i.e. 12pm The mean day is the time from one mean noon to the next

8. The mean Sun 2 (Local) mean time is UTC adjusted for longitude (Local) mean time is time by the mean sun (depends on longitude and the EoT) Standard time is the official time, including time zone and DST

9. Mean time and true time Make sure you understand the difference between “true time” and “time of true noon” Look at the following table: True time of true noon = True time of mean noon = Mean time of true noon = Mean time of mean noon =

10. Mean time and true time 2 Two of these are easy. True time of true noon = Mean time of mean noon = 12 Suppose the true Sun if 15 min early. In that case true time of mean noon = 12:15 while mean time of true noon = 11:45

11. Equation of time Equation of time = true time – mean time If evaluate both times at mean noon, we get 12:15 - 12 = 0:15 If we evaluate at true noon, we get 12 - 11:45 = 0:15

12. Equation of time 2 So if the true Sun is ahead, then: 1. EoT is positive 2. True time is later than mean time 3. The mean time of true noon is earlier than the mean time of mean noon

13. Equation of time 3 true time = mean time + EoT Used to be mean time = true time + EoT Now we know mean time, but not true time The EoT is positive when the true Sun is fast

14. When is the true day longest? The length of the true day depends on the speed of the eastward (yearly) motion of the Sun When the Sun moves quickly eastward, the westward motion is slower, and the true day is longer

15. Why does the eastward motion change? The are two reasons One is the obliquity of the Earth’s axis One is the eccentricity, i.e., the orbit of the Earth is an ellipse, not a circle We will consider them one at the time

16. Obliquity The ecliptic is not parallel to the celestial equator At the time of the solstices it is parallel to the celestial equator At the time of the equinoxes it is tilted to the celestial equator This makes the eastward motion faster around the solstices

17. Obliquity 2

18. Obliquity 3 This can also be seen as follows

19. Obliquity 4 Remember that when the true Sun is behind in the annual motion, it is ahead in the Sky

20. Obliquity 5 Early February is the time when the true Sun is the furthest behind. It is also the time when it goes from moving faster backwards to moving slower backwards.

21. Eccentricity When the Earth is at perihelion near January 4, the Earth will revolve faster, and the eastward motion of the Sun with respect to the stars is faster This makes the true day longer around perihelion

22. Eccentricity 2 The EoT is the cumulative effect When the Sun moves fastest/slowest eastward, the graph is steepest down/up

23. Combined effect The obliquity gives two fast periods The eccentricity gives one fast period The combined effect is one very fast period in December-January and one medium fast period in June-July

24. Combined effect 2

25. Combined effect 3

26. Extremes of equation of time maximum value (16m 25s) around November 3 (I will usually round this to 17m or 15m) minimum value (-14m 15s) around February 11 (I will usually round this to -14m or 15m)

27. Setting analemma

28. Rising and setting analemma

29. Rising analemma at the Equator

30. Earliest and latest sunrise in Singapore Singapore is at latitude 1N The sunrise time varies between 6:46 a.m. and 7:17 a.m., with the earliest sunrise on November 1 and the latest on February 9 In the same way, the sunset time varies between 6:50 p.m. and 7:21 p.m., with the earliest sunset on November 5 and the latest on February 13

31. Earliest and latest sunrise in Singapore 2 Singapore is at longitude about 103.75°E, so on average, the Sun will cross the meridian at 13:05 In Early November the Sun will therefore cross around 12:48 and in mid February around 13:19

32. Earliest and latest sunrise in Singapore 3 Because of refraction and the fact that sunrise and sunset are determined by the upper limb of the Sun and not the center of the Sun, the length of the day on the Equator is about 12h 8m At latitude 1N, the length of the day will vary by about 9 min, between 12h 3m and 12h 12m

33. Earliest and latest sunrise in Singapore 4 In both early November and mid February, the length of the day will be about 12h 04m, so if the Sun crosses the meridian at 13:19, the sunrise will be at 7:17 a.m. and the sunset at 7:21 p.m. If the Sun crosses the meridian at 12:48, the sunrise will be at 6:46 a.m. and the sunset will be 6:50 p.m.

34. Flowering time for plants in Singapore Some plants flower when the time of sunrise changes most rapidly Not based on earliest sunrise or longest day

35. Rising analemma at 5N Penang, Malaysia is at 5N

36. Rising analemma at 12N Bangkok is at 13N. There is a 50 days period between August 22 and October 10 when the variation in the sunrise in within an interval of 20 seconds!

37. Rising analemma at 25N

38. Rising analemma at 40N Beijing is at 40N

39. Rising analemma at 60N Oslo, Norway is at 60N

40. Rising southern analemma

41. Setting northern analemma

42. Setting southern analemma