A challenging geometric problem: Derive the sun path, the sun's trajectory in the sky, given a latitude, and the day of the year, e.g. 35° N, April 23.
I tried, and failed at first, to deduce accurately the shape of the sun paths. I then found sun path diagrams for specific latitudes here (for selected locations) and here* (for selected latitudes). See examples at the end of this post.
*carrying a wrong statement: "For a site located in the tropics between 23.5°N and 23.5°S, the sun will be in the North during the summer and in the South during the Winter." The correct statement is: "For a site located in the tropics between 23.5°N and 23.5°S, the sun will occupy only azimuths north of the E-W line (in the North) at the summer solstice and occupy only azimuths south of the E-W line (in the South) at the Winter solstice."
However, these sun path diagrams (linked above) do not reveal the simple geometry of the sun paths. Having fully thought through and understood the sun path geometry (I've not yet found a full description of this geometry online), I'll now briefly describe it (using some excellent diagrams from an excellent site):
(source)
These three diagrams show how sun paths can be readily determined. Note that in the 50° N (latitude) diagram, the angle 40° (of the noon sun at the equinoxes) is computed thus: 40°=90°-50°. In general, the angle of the noon sun (from the horizon) at the equinoxes equals (90° - latitude). Also note that the angle between the noon sun at the equinox and the noon sun at the (summer and winter) solstice is always 23.5°, whatever the latitude. You can therefore draw a similar diagram for any latitude. (For example, the sun paths at 40°N are as follows.)
Precisely stated, the sun paths at latitude L°N are formed by rotating the north polar (90°N) sun paths (shown above) by (90-L)°, clockwise when viewing from E to W, about the E-W axis.
The sun paths at latitude L°S are formed by rotating the north polar (90°N) sun paths by (90+L)°, clockwise when viewing from E to W, about the E-W axis.
More concisely (but using technical terms), when the sun's declination is d° (at a certain time of the year), the sun path is the d° small circle (parallel) of the celestial sphere. The latitude of the observer on earth determines the small circle's position in the sky (i.e. the degree of rotation from the north polar sun path).
I've omitted the proof of why the sun paths follow the simple geometry shown here, unless there is a reader's request for it.
[Hint for proof. First establish the north polar sun paths (as shown in the 90° N diagram). Then show that sun paths at any other latitude are formed by rotating (about axis E-W) the polar sun paths.] Without the hint, it would have been a challenging exercise to arrive at an elegant and succinct proof.
Sun's declination
To find out the sun's declination for any day of the year (+23.5° for the June solstice, 0° for the equinoxes, and -23.5° for the December solstice), you can use this table.
[Hint for proof. First establish the north polar sun paths (as shown in the 90° N diagram). Then show that sun paths at any other latitude are formed by rotating (about axis E-W) the polar sun paths.] Without the hint, it would have been a challenging exercise to arrive at an elegant and succinct proof.
Sun's declination
To find out the sun's declination for any day of the year (+23.5° for the June solstice, 0° for the equinoxes, and -23.5° for the December solstice), you can use this table.
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Star Paths
The reason for the sun paths' geometry in fact also applies to the trajectory across the sky of all relatively stationary celestial bodies, i.e. stars and the moon.
Note that when the sun's declination is +23.5° (for the June solstice), the sun path is the +23.5° (23.5°N) small circle (parallel) of the celestial sphere. In general, when the sun's declination is d° (at a certain time of the year), the sun path is the d° small circle of the celestial sphere. The latitude of the observer on earth determines the small circle's position in the sky (i.e. the degree of rotation from the north polar sun path).
This fact is also true of all stars. If a star's declination is d°, then its path across the sky is the d° small circle of the celestial sphere. The latitude of the observer on earth determines the small circle's position in the sky (i.e. the degree of rotation from the north polar star path).
The North Star (northern pole star)'s declination is +90°, and therefore appears stationary in the sky. All other stars appear to rotate around the northern and (imaginary) southern pole stars (currently there is no star at declination -90°).
Moon Path
Because the moon's orbital plane around the earth is close to the earth's orbital plane around the sun (the ecliptic), the moon's declination ranges from -23.5° to +23.5° (approximately; for precise range, see here) through a lunar cycle (a sidereal month of 27.32 days, slightly shorter than the period of moon's phases) (see diagram below).
Therefore the moon path is approximately within the range of the sun path, from -23.5° to +23.5° parallel.
Phases of the Moon
At any phase of the moon, the lit portion of the moon indicates the sun's position relative to the moon. The moon moves along a d° small circle of the celestial sphere, where d° is the moon's declination.
Thus, in the following table, the lunar phase determines the moon's meridian passing (upper culmination) time. The moon's declination and latitude determine the moonrise and moonset azimuth and the meridian passing altitude. The lunar phase, the moon's declination, and latitude determine the moonrise and moonset time.
For similar information on the moon path (and the sun path) at various locations, see here.
(table explanation) (
: southeast, 
: east, 
: southwest)
Star Paths
The reason for the sun paths' geometry in fact also applies to the trajectory across the sky of all relatively stationary celestial bodies, i.e. stars and the moon.
Note that when the sun's declination is +23.5° (for the June solstice), the sun path is the +23.5° (23.5°N) small circle (parallel) of the celestial sphere. In general, when the sun's declination is d° (at a certain time of the year), the sun path is the d° small circle of the celestial sphere. The latitude of the observer on earth determines the small circle's position in the sky (i.e. the degree of rotation from the north polar sun path).
This fact is also true of all stars. If a star's declination is d°, then its path across the sky is the d° small circle of the celestial sphere. The latitude of the observer on earth determines the small circle's position in the sky (i.e. the degree of rotation from the north polar star path).
The North Star (northern pole star)'s declination is +90°, and therefore appears stationary in the sky. All other stars appear to rotate around the northern and (imaginary) southern pole stars (currently there is no star at declination -90°).
Long exposure (45 min) photo (facing north) of the northern sky (50°N)
around North Star, showing the +40° to +90° small circles
(which are completely visible in the sky) of the celestial sphere (source)
Moon Path
Because the moon's orbital plane around the earth is close to the earth's orbital plane around the sun (the ecliptic), the moon's declination ranges from -23.5° to +23.5° (approximately; for precise range, see here) through a lunar cycle (a sidereal month of 27.32 days, slightly shorter than the period of moon's phases) (see diagram below).
Moon's declination, June 2012 (source)
Phases of the Moon
Appearance of the Moon at the North Pole. The upper part of the diagram is not to scale,
At the north pole, the moon's appearance is as shown above, and right (→) is the direction of the moon's advance along the small circle of the celestial sphere. The first quarter moon has the right half lit. The last quarter moon has the left half lit. The moon moves right along a celestial small circle.
Identically with the sun path and star paths, the moon's path (a celestial small circle) occupies a rotated position (from its north polar position) in the sky according to latitude. The first quarter moon's lit half always points to the moon's direction of advance through the night. Likewise the last quarter moon's dark half always points to the moon's direction of advance through the night. (see photos below, possibly taken from space)
At the equator, the first quarter moon rises with top half lit, and sets with the bottom half lit. The last quarter moon rises with the bottom half lit, and sets with the top half lit.
At the south pole the The first quarter moon has the left half lit. The last quarter moon has the right half lit. The moon moves left.
First quarter moon rising (noon, invisible),
or last quarter moon setting (noon, invisible), at the equator
First quarter moon setting (midnight),
or last quarter moon rising (midnight), at the equator
First quarter moon rising (around noon, invisible) at mid Northern hemisphere, or
last quarter moon setting (around noon, invisible) at mid Southern hemisphere.
First quarter moon setting (around midnight) at mid Northern hemisphere, or
last quarter moon rising (around midnight) at mid Southern hemisphere.
Last quarter moon rising (around midnight) at mid Northern hemisphere, or
first quarter moon setting (around midnight) at mid Southern hemisphere.
Last quarter moon setting (around noon, invisible) at mid Northern hemisphere, or
first quarter moon rising (around noon, invisible) at mid Southern hemisphere.
2012 Phases of the Moon
Universal Time
New Moon First Quarter Full Moon Last Quarter
d h m d h m d h m d h m
Jan 1 6 15 Jan 9 7 30 Jan 16 9 08
Jan 23 7 39 Jan 31 4 10 Feb 7 21 54 Feb 14 17 04
Feb 21 22 35 Mar 1 1 21 Mar 8 9 39 Mar 15 1 25
Mar 22 14 37 Mar 30 19 41 Apr 6 19 19 Apr 13 10 50
Apr 21 7 18 Apr 29 9 57 May 6 3 35 May 12 21 47
May 20 23 47 May 28 20 16 Jun 4 11 12 Jun 11 10 41
Jun 19 15 02 Jun 27 3 30 Jul 3 18 52 Jul 11 1 48
Jul 19 4 24 Jul 26 8 56 Aug 2 3 27 Aug 9 18 55
Aug 17 15 54 Aug 24 13 54 Aug 31 13 58 Sep 8 13 15
Sep 16 2 11 Sep 22 19 41 Sep 30 3 19 Oct 8 7 33
Oct 15 12 02 Oct 22 3 32 Oct 29 19 49 Nov 7 0 36
Nov 13 22 08 Nov 20 14 31 Nov 28 14 46 Dec 6 15 31
Dec 13 8 42 Dec 20 5 19 Dec 28 10 21
|
(source)
Rising and setting of the Moon
The sun is at its upper culmination (highest point in the sky), crossing the observer's meridian, at noon. The new moon is at its upper culmination also at noon (i.e. the moon is then between the sun and the earth). The moon culminates (at its highest point in the sky) at 3 pm at waxing crescent, 6 pm at first quarter, 12 midnight at full moon, and 6 am at last quarter. (see Lunar phase)
At the equator, the moon rises about 6 hours before culmination, and sets about 6 hours after culmination. Elsewhere, the declination of the moon and the observer's latitude determines the exact time of the moon's rising and setting.
Thus, in the following table, the lunar phase determines the moon's meridian passing (upper culmination) time. The moon's declination and latitude determine the moonrise and moonset azimuth and the meridian passing altitude. The lunar phase, the moon's declination, and latitude determine the moonrise and moonset time.
For similar information on the moon path (and the sun path) at various locations, see here.
Rising and setting times for the Moon. London, July 2012 (source)
All times are in local time for London (BST=UTC+1h)(table explanation) (
: southeast, : east, : southwest)| Time,local | Azimuth | Meridian Passing | |||||||
|---|---|---|---|---|---|---|---|---|---|
| Date | Moonrise | Moonset | Moonrise | Moonset | Time | Altitude | Distance | Illuminated | Phase |
| (km) | |||||||||
| 1 Jul 2012 | - 19:04 | 02:27 - | - 126° | 235° - | 23:14 | 15.9° | 362,389 | 95.1% | |
| 2 Jul 2012 | - 20:05 | 03:23 - | - 126° | 233° - | |||||
| 3 Jul 2012 | - 20:54 | 04:30 - | - 123°  | 235° - | 00:16 | 16.4° | 363,485 | 99.0% | Full Moon at 19:52 |
| 4 Jul 2012 | - 21:32 | 05:45 - | - 117° | 239° - | 01:16 | 18.4° | 366,204 | 99.8% | |
| 5 Jul 2012 | - 22:02 | 07:04 - | - 111° | 245° - | 02:14 | 21.7° | 370,353 | 97.7% | |
| 6 Jul 2012 | - 22:27 | 08:22 - | - 103° | 252° - | 03:07 | 25.9° | 375,566 | 92.9% | |
| 7 Jul 2012 | - 22:49 | 09:37 - | - 95° | 260° - | 03:56 | 30.6° | 381,369 | 86.1% | |
| 8 Jul 2012 | - 23:09 | 10:49 - | - 88° | 269° - | 04:43 | 35.5° | 387,254 | 77.7% | |
| 9 Jul 2012 | - 23:29 | 11:59 - | - 80° | 276° - | 05:28 | 40.4° | 392,751 | 68.3% | |
| 10 Jul 2012 | - 23:50 | 13:07 - | - 73°  | 284° - | 06:12 | 45.0° | 397,473 | 58.5% | |
| 11 Jul 2012 | 14:13 | 291° | 06:55 | 49.2° | 401,138 | 48.5% | Third Quarter at 02:48 | ||
| 12 Jul 2012 | 00:13 | 15:18 | 67° | 296° | 07:40 | 52.8° | 403,571 | 38.7% | |
| 13 Jul 2012 | 00:39 | 16:21 | 62° | 301° | 08:25 | 55.7° | 404,701 | 29.5% | |
| 14 Jul 2012 | 01:11 | 17:21 | 57° | 305° | 09:12 | 57.8° | 404,561 | 21.0% | |
| 15 Jul 2012 | 01:49 | 18:16 | 55° | 306° | 10:00 | 59.0° | 403,277 | 13.6% | |
| 16 Jul 2012 | 02:34 | 19:04 | 54° | 306° | 10:50 | 59.2° | 401,050 | 7.5% | |
| 17 Jul 2012 | 03:28 | 19:45 | 55° | 303° | 11:40 | 58.3° | 398,128 | 3.1% | |
| 18 Jul 2012 | 04:29 | 20:20 | 58° | 299° | 12:30 | 56.3° | 394,760 | 0.6% | |
| 19 Jul 2012 | 05:36 | 20:50 | 63° | 294° | 13:19 | 53.3° | 391,174 | 0.3% | New Moon at 05:25 |
| 20 Jul 2012 | 06:46 | 21:15 | 69° | 287° | 14:08 | 49.5° | 387,548 | 2.2% | |
| 21 Jul 2012 | 07:59 | 21:38 | 76° | 280° | 14:55 | 44.9° | 384,013 | 6.4% | |
| 22 Jul 2012 | 09:12 | 22:00 | 84° | 272° | 15:43 | 40.0° | 380,654 | 12.9% | |
| 23 Jul 2012 | 10:27 | 22:22 | 92° | 264° | 16:31 | 34.9° | 377,523 | 21.3% | |
| 24 Jul 2012 | 11:44 | 22:45 | 101° | 256° | 17:20 | 29.8° | 374,659 | 31.4% | |
| 25 Jul 2012 | 13:01 | 23:12 | 109° | 248° | 18:11 | 25.1° | 372,108 | 42.7% | |
| 26 Jul 2012 | 14:19 | 23:43 | 116° | 241° | 19:05 | 21.0° | 369,955 | 54.5% | First Quarter at 09:56 |
| 27 Jul 2012 | 15:36 | - | 122° | - | 20:02 | 18.0° | 368,335 | 66.2% | |
| 28 Jul 2012 | - 16:49 | 00:23 - | - 126° | 237° - | 21:01 | 16.3° | 367,431 | 77.1% | |
| 29 Jul 2012 | - 17:52 | 01:12 - | - 126° | 234° - | 22:02 | 16.1° | 367,439 | 86.4% | |
| 30 Jul 2012 | - 18:45 | 02:13 - | - 125° | 234° - | 23:02 | 17.4° | 368,522 | 93.5% | |
| 31 Jul 2012 | - 19:27 | 03:23 - | - 120° | 237° - | 23:59 | 20.0° | 370,752 | 98.1% | |
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Some sun path diagrams
Equator
London, UK (51.4°N)
Arctic circle
Singapore. 9 Feb 2011
Singapore, March equinox
Singapore, September equinox
Singapore, June solstice
Singapore, December solstice
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Hong Kong (22°19′N, near Tropic of Cancer), June solstice
Durham, UK. 8 Feb 2011
Sun path at Qanaq (Qaanaaq), Greenland (77°29′00″N, within the Arctic Circle) during soltices and equinoxes:
Summer solstice
Autumn equinox
Winter solstice
Spring equinox
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