The simple geometry of sun, moon, and star paths

How to determine the sun, moon, and stars' path across the sky at any latitude, and at any time of the year.



A challenging geometric problem: Derive the sun path (or day arc of the sun), 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:

Sun path geometry

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.)

sun paths at 40°N

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).


Why do the sun paths follow the simple geometry shown above?

Standing upright at θ° N

It is easy to see that, when you stand upright at the north pole, the sun paths are as shown in the 90° N diagram. 

Standing upright at any other latitude (as shown in the above diagram) causes the sun paths to rotate in the way described earlier.


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°).

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 a precise range, see here) through a lunar cycle (a sidereal month of  27.32 days, slightly shorter than the period of moon's phases (synodic month) of 29 days, 12 hours, 44 minutes) (see diagrams below).

Moon's declination, June 2012 (source)

Moon's declination, July 2012

Therefore the moon path is approximately within the range of the sun path, from -23.5° to +23.5° parallel.

For detailed data on lunar declination from 2001-2100, see here.

Lunar Phases and Appearance (Shape) of the Moon 

Appearance of the Moon at the North Pole. The upper part of the diagram is not to scale,

 as the Moon is much farther from the Earth than shown here.(source)

 

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.

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

Source: All the following diagrams are from here, where you can specify the location and date of the sun path you want.

Sun path at Qanaq (Qaanaaq), Greenland (77°29′00″N, above the Arctic Circle) at summer solstice

Sun path at Trondheim (63°25′N, just below the Arctic Circle) at summer solstice

Sun path at Hong Kong (22°19′N, near Tropic of Cancer) at summer solstice



Sun path at Quito (near the Equator, 0°13′S) at March equinox

Sun path at Quito (near the Equator, 0°13′S) at June solstice

Sun path at Bangkok, Thailand (13°55′N) at June solstice

Sun path at Bangkok, Thailand (13°55′N) at December solstice

Why does the setting waxing crescent moon appear more horizontal (seem to smile more) in winter than in summer in the Northern hemisphere?

A reader of this post has asked about the phenomenon that the waxing crescent moon (following the new moon), when observed setting in the west in mid-northern latitudes, appears to be more horizontal (like a smile) in winter than in summer.

A common, but wrong, interpretation is that the moon path in winter is different from that in summer, as shown in this diagram:

Wrong picture! 

There is absolutely no reason why the moon path should differ in this manner.

The moon path intercepts the western  horizon at exactly the same angle (given any fixed latitude) at all times of the year. 

The following is the right picture, showing that the crescent moon tilts but the moon path remains the same:

The explanation of this tilt is as follows:

For background knowledge, first read the section above on Lunar Phases and Appearance (Shape) of the Moon.

Appearance of the Moon at the North Pole

For the Moon to appear (called the standard appearance) as shown in the above diagram, a specific relation (called the special configuration) on the positions of the earth, the moon and the sun must hold.

You may skip the following description of the special configuration, the understanding of which requires some careful thought.

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Special Configuration

Consider an observer K of the Moon on Earth's North Pole. The boundary C of the Moon's observable hemisphere (assuming full moon) is a great circle. Let's impose circles of longitude (meridians) on the moon thus: Let the top and bottom of the Moon (on C) as seen by K be the Moon's poles. A Moon's meridian is a great circle, such as C, that contains both poles.

For the Moon to appear as shown above with the standard appearance, the boundary of the Moon's Sun-illuminated hemisphere must be a meridian.

This in turn implies that the Sun must lie on the Moon's Equatorial plane (the plane normal to its polar axis and containing its center). This is the special configuration mentioned above.

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Near the June solstice, the sun's declination is high (near +23.5°). Thus it is probable the the sun's declination is higher than what the special configuration requires, which in turn causes the moon shape as seen in the North Pole to tilt in the following manner compared with the standard appearance because the sun now illuminates a different portion of the moon.

Shape of first quarter moon (tilt exaggerated) at North Pole  

Near the December solstice, the sun's declination is low (near -23.5°). Thus it is probable the the sun's declination is lower than what the special configuration requires, which in turn causes the moon shape as seen in the North Pole to tilt in the following manner compared with the standard appearance because the sun now illuminates a different portion of the moon.

Shape of first quarter moon (tilt exaggerated) at North Pole

The moon shape seen at any latitude likewise tilts (see the section above on Lunar Phases and Appearance (Shape) of the Moon for how latitude determines moon shape).

This explains why the waxing crescent moon, when setting in the northern hemisphere, probably smiles more in winter than in summer.

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The following table gives the approximate data on the Moon's declination, the Sun's declination as required by the special configuration (A), the Sun's actual declination (B), and their difference (B-A), two days after the new moons (i.e. at the first observable waxing crescent moon) in 2015. 

The sun's deviation from its special configuration is 11.49° on 7/18/2015, and -13.94° on 2/21/2015, making a total deviation of 25.43°.  This is certainly noticeable, and explains the above mentioned shift in moon shape.

Date	moon's declination	A. special config	B. sun's declination	B-A
1/22/2015	-8.40	-7.65	-19.87	-12.22
2/21/2015	3.37	3.07	-10.87	-13.94
3/22/2015	9.98	9.08	0.40	-8.68
4/21/2015	17.15	15.58	11.65	-3.93
5/20/2015	18.38	16.69	19.85	3.16
6/18/2015	17.08	15.51	23.40	7.89
7/18/2015	10.60	9.64	21.13	11.49
8/16/2015	4.83	4.40	13.93	9.53
9/15/2015	-5.13	-4.67	3.27	7.94
10/15/2015	-13.73	-12.48	-8.30	4.18
11/14/2015	-18.20	-16.53	-18.07	-1.54
12/13/2015	-17.77	-16.14	-23.12	-6.98

The sun's pattern of deviation from its special configuration varies from year to year. The maximum declination of the moon plays an important role in the pattern.

October 2015 sees a minor lunar standstill, when the moon's maximum declination is at its minimum.

The following tabulates the same set of data for 2006. In June 2006 there was a major lunar standstill, when the moon's maximum declination is at its maximum.

Date	moon's declination	A. special config	B. sun's declination	B-A
1/2/2006	-24.23	-21.95	-22.85	-0.90
1/31/2006	-18.12	-16.45	-17.63	-1.18
3/2/2006	-6.33	-5.76	-7.87	-2.11
3/31/2006	7.57	6.89	3.62	-3.27
4/30/2006	20.50	18.60	14.20	-4.40
5/29/2006	28.49	25.75	21.50	-4.25
6/28/2006	24.52	22.21	23.12	0.91
7/27/2006	19.90	18.06	19.45	1.39
8/26/2006	9.53	8.67	11.07	2.40
9/24/2006	-3.85	-3.51	0.03	3.54
10/24/2006	-17.50	-15.89	-11.22	4.67
11/23/2006	-28.44	-25.70	-20.03	5.67
12/22/2006	-24.52	-22.21	-23.50	-1.29

The largest positive deviation of the sun's declination (5.67°) in 2006 occurred near the December solstice, and the largest negative deviation of the sun's declination (-4.40°) in 2006 occurred near the June solstice. The magnitude of the deviations were small throughout the year. 

The above mentioned shift in crescent moon shape is therefore not an unchanging phenomenon observable every year.


* For computing column A. above, I used the geometry and trigonometry here.

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