Home·Sun angle and daylight

224 sun positions — at 40°N the June solstice runs 14 h 51 min of daylight

Every pairing of 14 latitudes and 16 dates, worked out as a noon altitude, a length of day, sunrise and sunset in solar time, and a shadow length. Nothing is copied from a table: the day of the year gives the declination, and the declination meets the latitude. Refraction and the equation of time are left out, so these sunrise times sit several minutes — sometimes a quarter of an hour — away from clock time.

The declination comes from the date alone

The earth’s axis leans 23.44° out of its orbital plane, so the latitude that has the sun directly overhead at noon travels between −23.44° and +23.44° over a year. That latitude is the declination. The formula in common use is δ = 23.44° × sin(360° × (284 + N) ÷ 365), where N counts the days from 1 January. It reaches +23.44° at the June solstice, −23.44° at the December solstice, and crosses zero at the equinoxes — though this approximation puts the crossing about two days late.

Latitude and declination together fix the length of day

The hour angle H at which the sun meets the horizon follows from cos H = −tan(latitude) × tan(declination). The sky turns 15° an hour, so the day lasts 2H ÷ 15 hours, sunrise falls H ÷ 15 hours before noon and sunset the same amount after. When |cos H| passes 1 the arithmetic has not broken — that is the answer: above 1 the sun never rises (polar night), below −1 it never sets (midnight sun). Both show up in this table at 70°N in June and December, and those cells carry no clock times.

A shadow is the tangent of the altitude, read backwards

If the noon altitude is h, a pole 1 m tall casts 1 ÷ tan h metres of shadow. Measure the shadow instead and the altitude falls out, and from the altitude you can work back to a date or a latitude — that is how Eratosthenes measured the earth in antiquity. Shadows stretch fast as the sun drops: at 45° the shadow equals the height, at 30° it is 1.7 times as long, and at 5° more than eleven times.

What this leaves out

Sunrise here means the centre of the sun crossing the geometric horizon, an altitude of 0°. The usual convention adds refraction and the sun’s own radius and uses −0.833°, which makes the day 8 to 10 minutes longer at mid-latitudes. The clock is apparent solar time: to reach civil time you still have to add the equation of time (±16 minutes), the gap between your longitude and your time zone, and any summer time. The declination formula carries an error of up to 1.5°, leap years are not counted, so dates after February sit one day out, and neither the observer’s elevation nor hills and buildings on the horizon are in here.

A high sun is a strong sun

The higher the sun, the less air its light passes through, and the stronger the ultraviolet. As a rule of thumb, when your shadow is shorter than you are, the ultraviolet is near its peak for the day — how long skin lasts before it reddens depends on the index and the skin type.

See burn times by UV index

At a glance

Length of day · Noon altitude

60°SJune solstice 6.56° · December solstice 53.44°

January 15
17 h 39 min51.26°
February 15
15 h 13 min43.28°
March 15
12 h 39 min32.82°
March 20
12 h 11 min30.81°
April 15
9 h 47 min20.59°
May 15
7 h 11 min11.22°
June 15
5 h 34 min6.7°
June 21
5 h 31 min6.56°
July 15
6 h 16 min8.49°
August 15
8 h 39 min16.22°
September 15
11 h 29 min27.78°
September 23
12 h 14 min31.01°
October 15
14 h 16 min39.6°
November 15
16 h 56 min49.14°
December 15
18 h 27 min53.33°
December 21
18 h 29 min53.44°

50°SJune solstice 16.56° · December solstice 63.44°

January 15
15 h 41 min61.26°
February 15
14 h 11 min53.28°
March 15
12 h 27 min42.82°
March 20
12 h 8 min40.81°
April 15
10 h 29 min30.59°
May 15
8 h 49 min21.22°
June 15
7 h 53 min16.7°
June 21
7 h 51 min16.56°
July 15
8 h 16 min18.49°
August 15
9 h 44 min26.22°
September 15
11 h 39 min37.78°
September 23
12 h 10 min41.01°
October 15
13 h 33 min49.6°
November 15
15 h 15 min59.14°
December 15
16 h 7 min63.33°
December 21
16 h 9 min63.44°

40°SJune solstice 26.56° · December solstice 73.44°

January 15
14 h 32 min71.26°
February 15
13 h 31 min63.28°
March 15
12 h 19 min52.82°
March 20
12 h 5 min50.81°
April 15
10 h 56 min40.59°
May 15
9 h 47 min31.22°
June 15
9 h 10 min26.7°
June 21
9 h 9 min26.56°
July 15
9 h 26 min28.49°
August 15
10 h 25 min36.22°
September 15
11 h 45 min47.78°
September 23
12 h 7 min51.01°
October 15
13 h 5 min59.6°
November 15
14 h 15 min69.14°
December 15
14 h 50 min73.33°
December 21
14 h 51 min73.44°

30°SJune solstice 36.56° · December solstice 83.44°

January 15
13 h 44 min81.26°
February 15
13 h 3 min73.28°
March 15
12 h 13 min62.82°
March 20
12 h 4 min60.81°
April 15
11 h 16 min50.59°
May 15
10 h 29 min41.22°
June 15
10 h 5 min36.7°
June 21
10 h 4 min36.56°
July 15
10 h 15 min38.49°
August 15
10 h 55 min46.22°
September 15
11 h 50 min57.78°
September 23
12 h 5 min61.01°
October 15
12 h 45 min69.6°
November 15
13 h 32 min79.14°
December 15
13 h 55 min83.33°
December 21
13 h 56 min83.44°

20°SJune solstice 46.56° · December solstice 86.56°

January 15
13 h 5 min88.74°
February 15
12 h 39 min83.28°
March 15
12 h 8 min72.82°
March 20
12 h 2 min70.81°
April 15
11 h 32 min60.59°
May 15
11 h 3 min51.22°
June 15
10 h 48 min46.7°
June 21
10 h 47 min46.56°
July 15
10 h 54 min48.49°
August 15
11 h 19 min56.22°
September 15
11 h 54 min67.78°
September 23
12 h 3 min71.01°
October 15
12 h 28 min79.6°
November 15
12 h 58 min89.14°
December 15
13 h 12 min86.67°
December 21
13 h 13 min86.56°

10°SJune solstice 56.56° · December solstice 76.56°

January 15
12 h 31 min78.74°
February 15
12 h 19 min86.72°
March 15
12 h 4 min82.82°
March 20
12 h 1 min80.81°
April 15
11 h 47 min70.59°
May 15
11 h 32 min61.22°
June 15
11 h 25 min56.7°
June 21
11 h 25 min56.56°
July 15
11 h 28 min58.49°
August 15
11 h 40 min66.22°
September 15
11 h 57 min77.78°
September 23
12 h 1 min81.01°
October 15
12 h 14 min89.6°
November 15
12 h 28 min80.86°
December 15
12 h 35 min76.67°
December 21
12 h 35 min76.56°

EquatorJune solstice 66.56° · December solstice 66.56°

January 15
12 h 0 min68.74°
February 15
12 h 0 min76.72°
March 15
12 h 0 min87.18°
March 20
12 h 0 min89.19°
April 15
12 h 0 min80.59°
May 15
12 h 0 min71.22°
June 15
12 h 0 min66.7°
June 21
12 h 0 min66.56°
July 15
12 h 0 min68.49°
August 15
12 h 0 min76.22°
September 15
12 h 0 min87.78°
September 23
12 h 0 min88.99°
October 15
12 h 0 min80.4°
November 15
12 h 0 min70.86°
December 15
12 h 0 min66.67°
December 21
12 h 0 min66.56°

10°NJune solstice 76.56° · December solstice 56.56°

January 15
11 h 29 min58.74°
February 15
11 h 41 min66.72°
March 15
11 h 56 min77.18°
March 20
11 h 59 min79.19°
April 15
12 h 13 min89.41°
May 15
12 h 28 min81.22°
June 15
12 h 35 min76.7°
June 21
12 h 35 min76.56°
July 15
12 h 32 min78.49°
August 15
12 h 20 min86.22°
September 15
12 h 3 min82.22°
September 23
11 h 59 min78.99°
October 15
11 h 46 min70.4°
November 15
11 h 32 min60.86°
December 15
11 h 25 min56.67°
December 21
11 h 25 min56.56°

20°NJune solstice 86.56° · December solstice 46.56°

January 15
10 h 55 min48.74°
February 15
11 h 21 min56.72°
March 15
11 h 52 min67.18°
March 20
11 h 58 min69.19°
April 15
12 h 28 min79.41°
May 15
12 h 57 min88.78°
June 15
13 h 12 min86.7°
June 21
13 h 13 min86.56°
July 15
13 h 6 min88.49°
August 15
12 h 41 min83.78°
September 15
12 h 6 min72.22°
September 23
11 h 57 min68.99°
October 15
11 h 32 min60.4°
November 15
11 h 2 min50.86°
December 15
10 h 48 min46.67°
December 21
10 h 47 min46.56°

30°NJune solstice 83.44° · December solstice 36.56°

January 15
10 h 16 min38.74°
February 15
10 h 57 min46.72°
March 15
11 h 47 min57.18°
March 20
11 h 56 min59.19°
April 15
12 h 44 min69.41°
May 15
13 h 31 min78.78°
June 15
13 h 55 min83.3°
June 21
13 h 56 min83.44°
July 15
13 h 45 min81.51°
August 15
13 h 5 min73.78°
September 15
12 h 10 min62.22°
September 23
11 h 55 min58.99°
October 15
11 h 15 min50.4°
November 15
10 h 28 min40.86°
December 15
10 h 5 min36.67°
December 21
10 h 4 min36.56°

40°NJune solstice 73.44° · December solstice 26.56°

January 15
9 h 28 min28.74°
February 15
10 h 29 min36.72°
March 15
11 h 41 min47.18°
March 20
11 h 55 min49.19°
April 15
13 h 4 min59.41°
May 15
14 h 13 min68.78°
June 15
14 h 50 min73.3°
June 21
14 h 51 min73.44°
July 15
14 h 34 min71.51°
August 15
13 h 35 min63.78°
September 15
12 h 15 min52.22°
September 23
11 h 53 min48.99°
October 15
10 h 55 min40.4°
November 15
9 h 45 min30.86°
December 15
9 h 10 min26.67°
December 21
9 h 9 min26.56°

50°NJune solstice 63.44° · December solstice 16.56°

January 15
8 h 19 min18.74°
February 15
9 h 49 min26.72°
March 15
11 h 33 min37.18°
March 20
11 h 52 min39.19°
April 15
13 h 31 min49.41°
May 15
15 h 11 min58.78°
June 15
16 h 7 min63.3°
June 21
16 h 9 min63.44°
July 15
15 h 44 min61.51°
August 15
14 h 16 min53.78°
September 15
12 h 21 min42.22°
September 23
11 h 50 min38.99°
October 15
10 h 27 min30.4°
November 15
8 h 45 min20.86°
December 15
7 h 53 min16.67°
December 21
7 h 51 min16.56°

60°NJune solstice 53.44° · December solstice 6.56°

January 15
6 h 21 min8.74°
February 15
8 h 47 min16.72°
March 15
11 h 21 min27.18°
March 20
11 h 49 min29.19°
April 15
14 h 13 min39.41°
May 15
16 h 49 min48.78°
June 15
18 h 26 min53.3°
June 21
18 h 29 min53.44°
July 15
17 h 44 min51.51°
August 15
15 h 21 min43.78°
September 15
12 h 31 min32.22°
September 23
11 h 46 min28.99°
October 15
9 h 44 min20.4°
November 15
7 h 4 min10.86°
December 15
5 h 33 min6.67°
December 21
5 h 31 min6.56°

70°NJune solstice 43.44° · December solstice -3.44°

January 15
0 h 0 min (Polar night)-1.26°
February 15
6 h 36 min6.72°
March 15
10 h 58 min17.18°
March 20
11 h 42 min19.19°
April 15
15 h 37 min29.41°
May 15
21 h 13 min38.78°
June 15
24 h 0 min (Midnight sun)43.3°
June 21
24 h 0 min (Midnight sun)43.44°
July 15
24 h 0 min (Midnight sun)41.51°
August 15
17 h 39 min33.78°
September 15
12 h 49 min22.22°
September 23
11 h 38 min18.99°
October 15
8 h 19 min10.4°
November 15
2 h 20 min0.86°
December 15
0 h 0 min (Polar night)-3.33°
December 21
0 h 0 min (Polar night)-3.44°

Worth knowing

  • Declination δ = 23.44° × sin(360° × (284 + N) ÷ 365), where N counts the days from 1 January.
  • Noon altitude = 90° − |latitude − declination|; at the June solstice that is 90° − φ + 23.44° in the north.
  • Length of day = 2H ÷ 15 hours with cos H = −tan(latitude) × tan(declination); beyond |cos H| = 1 lies polar day or night.
  • Shadow = height ÷ tan(altitude). Times are apparent solar time, with refraction and the equation of time left out.

Times are apparent solar time, with noon set to 12:00 when the sun crosses the meridian.

Common questions

Q. How do I work out the length of a day?

Take cos H = −tan(latitude) × tan(declination) to get the hour angle H, then the day lasts 2H ÷ 15 hours. The declination comes from the date, so latitude and date are all you need. At the equinoxes the declination is near zero and every latitude gets close to twelve hours.

Q. Why does the equator always get twelve hours?

At latitude 0 the tangent of the latitude is 0, so cos H = 0 whatever the declination, which puts H at 90°. The seasons come and go but the day barely moves from twelve hours — add refraction and it runs a few minutes longer.

Q. Can I set my clock by these sunrise times?

Not directly. They are apparent solar time, so you still have to add the equation of time (±16 minutes), the gap between your longitude and your time zone, and summer time. Refraction is missing too, which makes the real sunrise 4 to 5 minutes earlier than shown.