Wednesday, May 2, 2012

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 (using some diagrams from an excellent site):


Sun path geometry (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.)


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).


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 north 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.



*******************************************************************************

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.


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                
 


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: southeast, East: east,  Southwest: southwest)

Time,localAzimuthMeridian Passing
DateMoonriseMoonsetMoonriseMoonsetTimeAltitudeDistanceIlluminatedPhase
(km)
1 Jul 2012-
19:04
02:27
-
-
126°Southeast
235°Southwest
-
23:1415.9° 362,38995.1%
2 Jul 2012-
20:05
03:23
-
-
126°Southeast
233°Southwest
-
3 Jul 2012-
20:54
04:30
-
-
123°East-southeast
235°Southwest
-
00:1616.4° 363,48599.0%Full Moon at 19:52
4 Jul 2012-
21:32
05:45
-
-
117°East-southeast
239°West-southwest
-
01:1618.4° 366,20499.8%
5 Jul 2012-
22:02
07:04
-
-
111°East-southeast
245°West-southwest
-
02:1421.7° 370,35397.7%
6 Jul 2012-
22:27
08:22
-
-
103°East-southeast
252°West-southwest
-
03:0725.9° 375,56692.9%
7 Jul 2012-
22:49
09:37
-
-
95°East
260°West
-
03:5630.6° 381,36986.1%
8 Jul 2012-
23:09
10:49
-
-
88°East
269°West
-
04:4335.5° 387,25477.7%
9 Jul 2012-
23:29
11:59
-
-
80°East
276°West
-
05:2840.4° 392,75168.3%
10 Jul 2012-
23:50
13:07
-
-
73°East-northeast
284°West-northwest
-
06:1245.0° 397,47358.5%
11 Jul 201214:13291°West-northwest06:5549.2° 401,13848.5%Third Quarter at 02:48
12 Jul 201200:1315:1867°East-northeast296°West-northwest07:4052.8° 403,57138.7%
13 Jul 201200:3916:2162°East-northeast301°West-northwest08:2555.7° 404,70129.5%
14 Jul 201201:1117:2157°East-northeast305°Northwest09:1257.8° 404,56121.0%
15 Jul 201201:4918:1655°Northeast306°Northwest10:0059.0° 403,27713.6%
16 Jul 201202:3419:0454°Northeast306°Northwest10:5059.2° 401,0507.5%
17 Jul 201203:2819:4555°Northeast303°West-northwest11:4058.3° 398,1283.1%
18 Jul 201204:2920:2058°East-northeast299°West-northwest12:3056.3° 394,7600.6%
19 Jul 201205:3620:5063°East-northeast294°West-northwest13:1953.3° 391,1740.3%New Moon at 05:25
20 Jul 201206:4621:1569°East-northeast287°West-northwest14:0849.5° 387,5482.2%
21 Jul 201207:5921:3876°East-northeast280°West14:5544.9° 384,0136.4%
22 Jul 201209:1222:0084°East272°West15:4340.0° 380,65412.9%
23 Jul 201210:2722:2292°East264°West16:3134.9° 377,52321.3%
24 Jul 201211:4422:45101°East256°West-southwest17:2029.8° 374,65931.4%
25 Jul 201213:0123:12109°East-southeast248°West-southwest18:1125.1° 372,10842.7%
26 Jul 201214:1923:43116°East-southeast241°West-southwest19:0521.0° 369,95554.5%First Quarter at 09:56
27 Jul 201215:36-122°East-southeast-20:0218.0° 368,33566.2%
28 Jul 2012-
16:49
00:23
-
-
126°Southeast
237°West-southwest
-
21:0116.3° 367,43177.1%
29 Jul 2012-
17:52
01:12
-
-
126°Southeast
234°Southwest
-
22:0216.1° 367,43986.4%
30 Jul 2012-
18:45
02:13
-
-
125°Southeast
234°Southwest
-
23:0217.4° 368,52293.5%
31 Jul 2012-
19:27
03:23
-
-
120°East-southeast
237°West-southwest
-
23:5920.0° 370,75298.1%

******************************************************
Some sun path diagrams


Equator



London, UK (51.4°N)



Arctic circle


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 (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 key fact that explains this phenomenon is that the ecliptic (the earth's orbital plane around the sun) is slightly different from the moon's orbital plane around the earth (see this diagram below).

The sun is rarely on the moon's orbital plane
(For details, read this)

The sun lies exactly on the moon's orbital plane (around the earth) only twice a year.

Between these two points in time, the sun spends half a year (around the northern summer) above (as viewed from the North Pole) the moon's orbital plane, and half a year (around the northern winter) below (as viewed from the North Pole) the moon's orbital plane

Whenever the sun is above (as viewed from the North Pole) the moon's orbital plane, the moon's illuminated portion will be tilted slightly towards the exterior of the circle (there is an exterior of the circle, except when you are at the equator or the poles) formed by the moon path, when viewed in the northern hemisphere. This happens in the northern summer. (The tilt is towards the interior of the circle when viewed in the southern hemisphere.)

Whenever the sun is below (as viewed from the North Pole) the moon's orbital plane, the moon's illuminated portion will be tilted slightly towards the interior of the circle (there is an interior of the circle, except at the equator and the poles) formed by the moon path, when viewed in the northern hemisphere. This happens in the northern winter. (The tilt is towards the exterior of the circle when viewed in the southern hemisphere.)


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

Note. The above statements that in the northern summer, the sun is above (as viewed from the North Pole) the moon's orbital plane, and that in the northern winter, the sun is below (as viewed from the North Pole) the moon's orbital plane, hold true for the 9 years centering on a minor lunar standstill (one of which is in October 2015).  The statements therefore hold true from April 2011 to March 2020, approximately.

During the 9 years centering on a major lunar standstill (one of which is in April 2025), the following holds: In the northern summer, the sun is below (as viewed from the North Pole) the moon's orbital plane, and that in the northern winter, the sun is above (as viewed from the North Pole) the moon's orbital plane.

(For lunar standstill, see here.)




   *******************************************************************

14 comments:

  1. Thank you so much for this stunning exposition!
    I have come upon only now and I am going to study it and master the principles, so that I will be able to set up all my solar-powered contraptions and inventions and also have the ability to help other people with similar aspirations.

    ReplyDelete
  2. This comment has been removed by the author.

    ReplyDelete
  3. Hi, about to study it for my own purpose: Sunshade. I live in bangkok, where sunshade are put above kids instead of between the kids and the sun. So sometime they just don;t work when time is needed. If you have any idea, please help.

    ReplyDelete
    Replies
    1. I would first find out the lowest altitude of the sun during the year in the directions (azimuth) where I want shielding.

      Examine the sun path diagrams for Bangkok (which I have newly added to the end of my blogpost above) at the solstices (when the sun path is at the extremity of its range). [Visit http://sunposition.info/sunposition/spc/locations.php#1 for more.]

      Take 45° azimuth (NE) as an example. The lowest altitude of the sun at 45° azimuth is 76° (during the June solstice). Thus if the sun shade's altitude (measured from the spot to be shaded) in the 45° azimuth is 70° or less, then the shade works well in the 45° azimuth.

      I would work out the required altitude in all the azimuths needed.

      The azimuths 66°-114° (East) and 247°-295° (West) cannot be fully shaded as the sun rises and sets in these azimuths sometime during the year.

      Delete
  4. being as EVERY PERSON on this planet from rise to set looking twice in 12 hours seeing the same thing the moon flipping 180's at allll latitudes every 12 hours, this field rotation is out of date and not needed anymore for dazzthecameraman has made 6-7 videos using my name in the title tyo express how normal it is for field rotation to flip the moon regardless latitude, so either he is right or your right, im going to have to side with you, this is not normal,can we get a reading on how much of the extra 153 degree angular shift i am seeing is earth tilting on its axis?

    ReplyDelete
  5. Please provide the same data for the Southern Hemisphere.

    ReplyDelete
  6. I've provided links to the sources for the data in this post. I think if you look up the sources, you would find the data for locations in the Southern Hemisphere. However, if this is not true, then please specify which data, and I will find the relevant source.

    ReplyDelete
  7. The sun path diagrams at the end of the post are from here: http://sunposition.info/sunposition/spc/locations.php#1

    ReplyDelete
  8. Excellent article. Can you clarify the exact reason why the sunarcs are different in relation to the western horizon when the sun sets based on the time of year? This is empirically the case and I have seen it written. But do not understand visually why this is. It seems that during the vernal equinox the angle of the sun setting with the horizon is close to 90 degress (for an observer in the Northern hemisphere say at 40N) and much more acute at the autumnal equinox. Please explain why exactly this is so. Much appreciated. Thanks!

    ReplyDelete
    Replies
    1. Thank you. I am not aware of the phenomenon that you described, and cannot explain why there should be any difference between the sunpaths at the spring and the autumn equinoxes.

      Delete
  9. Hi. Thanks. Please see here under post from March 31, 2014 to see what I am referring to.

    http://astrobob.areavoices.com/tag/ecliptic/

    ReplyDelete
    Replies
    1. I believe that the moon path at the vernal equinox is the same as that at the autumnal equinox, but the sun illuminates a slightly different hemisphere of the moon (when viewed from Earth) at these times. This explains the different positions of the moon's visible crescent relative to the moon path, and therefore to the horizon.

      This is in turn due to the fact that the moon's orbital plane is not exactly the ecliptic.

      That diagram here ( http://en.m.wikipedia.org/wiki/Lunar_standstill#Detailed_explanation ) might clarify this fact.

      Delete
  10. I'm still confused as to the relationship between the moon path in a given season vs.the sun path. I can't see anywhere where the moon path would be affected by the season of the year (i.e. the point at which the sun is on the ecliptic).

    It seems that the only affects on the moon path are:

    1) the declination which is affected by the 18.6 year nodal cycle and
    2) the latitude of the observer (let's assume a mid latitude on the Northern Hemisphere).

    Is there any intuitive reason that the setting angle of the new moon would be at an acute angle with the horizon every spring and be close to vertical every fall? That is what many claim empirically and what is shown here:

    http://astrobob.areavoices.com/tag/ecliptic/#sthash.o9SOqscF.dpuf

    Thanks!


    Thanks!

    ReplyDelete
    Replies
    1. I have just added a section towards the end of this blogpost to explain in more depth. Please let me know if and where it isn't clear. Thanks.

      Delete