The sun’s height above the horizon follows from three angles: latitude, the sun’s declination for that day (between about -23.4 and +23.4 degrees) and the hour angle (15 degrees for every hour from solar noon). With sin(altitude) = sin(latitude) x sin(declination) + cos(latitude) x cos(declination) x cos(hour angle), 40 degrees north on 21 December at 10:00 solar time gives an altitude of 20.7 degrees.
Part of Passive Solar Building Design
The spherical-geometry relations that give the sun’s altitude, bearing, sunrise point and day length at any latitude, date and hour.
Three Angles Place the Sun
Latitude (positive north, negative south). Declination is how far north or south of the equator the sun is overhead on a given day. It swings between +23.44 degrees at the northern summer solstice and -23.44 degrees at the northern winter solstice. Hour angle h is the sun’s angular distance from the meridian: 0 at solar noon, 15 degrees per hour, negative in the morning and positive in the afternoon.
This page uses latitude negative in the Southern Hemisphere, so the same equations work in both hemispheres; the signs do the work. Dates quoted are Northern Hemisphere calendar dates. “Solar time” is time measured by the sun, which differs from clock time; see Finding True South for converting.
The Relations
Declination (Cooper’s equation, as given in Duffie and Beckman, Solar Engineering of Thermal Processes), with n the day of the year (1 January = 1):
declination = 23.45 degrees x sin(360 x (284 + n) / 365)
Altitude a:
sin a = sin(latitude) x sin(declination) + cos(latitude) x cos(declination) x cos h
Azimuth A (measured clockwise from true north), found from cos A = (sin(declination) - sin a x sin(latitude)) / (cos a x cos(latitude)), taking A as the arccosine for the morning and 360 degrees minus that for the afternoon.
Noon altitude: 90 degrees minus the absolute value of (latitude - declination). This is the carrier’s formula in general form.
Sunrise azimuth: cos A = sin(declination) / cos(latitude). Day length in hours: 2 x arccos(-tan(latitude) x tan(declination)) / 15.
How Accurate Is the Declination Formula?
Cooper’s equation is an approximation. Compared with the Fourier series of Spencer (1971) over every day of the year, computed for this page, the two differ by up to 1.4 degrees, with the largest gaps near the equinoxes (23 September: -1.0 degrees by Cooper’s equation, +0.25 degrees by Spencer’s). For a building layout good to a few degrees that is ample. It matters if you try to reproduce the noon altitude on a specific date to better than a degree. Declination values (Cooper’s equation) on the 21st of each month: January -20.1, February -11.2, March -0.4, April +11.6, May +20.1, June +23.4, July +20.4, August +11.8, September -0.2, October -11.8, November -20.4, December -23.4 degrees. The December and June values follow from the 23.44-degree tilt of the Earth’s axis.
Worked Example 1: 40 N, 21 December, 10:00 Solar Time
Latitude 40, declination -23.44, hour angle h = (10 - 12) x 15 = -30 degrees.
- sin a = sin 40 x sin(-23.44) + cos 40 x cos(-23.44) x cos(-30)
- = 0.6428 x (-0.3977) + 0.7660 x 0.9175 x 0.8660
- = -0.2557 + 0.6086 = 0.3529, so a = 20.7 degrees.
- Azimuth: cos A = (sin(-23.44) - sin 20.7 x sin 40) / (cos 20.7 x cos 40) = (-0.3977 - 0.3535 x 0.6428) / (0.9354 x 0.7660) = -0.6249 / 0.7165 = -0.8721, so A is about 151 degrees (morning, so no subtraction). The sun is south-south-east.
At solar noon on that day the altitude is 90 - |40 - (-23.44)| = 90 - 63.44 = 26.6 degrees, the same as the carrier’s 90 - 40 - 23.5. Three hours after noon (h = 45 degrees) the altitude is 14.0 degrees at azimuth 222 degrees.
Worked Example 2: 35 S
Latitude -35. On 21 June (southern winter) declination is +23.44.
- Noon altitude = 90 - |-35 - 23.44| = 90 - 58.44 = 31.6 degrees, with the sun due north (azimuth 0 degrees).
- Sunrise azimuth: cos A = sin 23.44 / cos 35 = 0.3977 / 0.8192 = 0.4855, A = 60.9 degrees: north-east.
- Day length = 2 x arccos(-tan(-35) x tan 23.44) / 15 = 2 x arccos(0.7002 x 0.4330) / 15 = 2 x arccos(0.3032) / 15 = 2 x 72.4 / 15 = 9.6 hours.
On 21 December at 35 S: declination -23.44, noon altitude 78.4 degrees, sunrise azimuth 119.1 degrees (south-east), day length 14.4 hours. The winter sun at 35 S is 31.6 degrees high at noon, in the north: the mirror of the Northern Hemisphere case.
Check for 40 N for comparison: sunrise azimuth on 21 June is 58.7 degrees (north-east) with a day length of 14.8 hours; on 21 December it is 121.3 degrees (south-east) with 9.2 hours.
A Field Check Without Calculation
At any time, the altitude a satisfies tan a = stick height / shadow length. The carrier’s quick check is the noon altitude formula. A stick and a tape measure replace the whole calculation if you can wait for the date; the arithmetic is for planning ahead.
Common Mistakes
- Sign of latitude in the southern hemisphere — negative, or the sun appears on the wrong side.
- Degrees against radians — set the calculator or table to degrees.
- Using clock time for hour angle — use solar time, 15 degrees per hour from solar noon.
- Using the 21st for the equinox — the equinoxes fall about 20 March and 22-23 September.
Safety
Measure by shadow, never by looking at the sun
The angles on this page can all be checked with a stick and its shadow. Do not point any sighting device at the sun.
What’s Next
- Learn next: Overhang Design for Seasonal Control — turning altitude into shade
- Combine with: Trigonometry Basics — the sine and cosine used here