Solar Declination Calculator
Solar declination is the angle between the Sun's rays and Earth's equatorial plane on a given day. It swings between about plus and minus 23.44 degrees through the year because Earth's axis is tilted, driving the seasons, the changing length of day, and the Sun's noon height. This calculator uses the widely used Cooper approximation to estimate declination from the day of the year. It also estimates the Sun's noon elevation for a latitude you enter. The result is accurate to about a degree, ideal for solar geometry and daylight planning rather than precise astronomy.
Solar declination formula
declination = 23.45 * sin(360 * (284 + N) / 365) (degrees)
N = day of year
noon elevation = 90 - |latitude - declination|
zenith angle = 90 - noon elevation
optimal noon tilt = latitude - declination
The Cooper formula traces the seasonal sine wave of declination. The Sun's noon elevation follows from latitude and declination; the optimal panel tilt at solar noon points the panel straight at the Sun.
Declination notes
- The default day 172 is near the June solstice (declination about plus 23.4 degrees).
- Declination is zero at the equinoxes and extreme at the solstices.
- The Cooper approximation is accurate to about one degree.
- Use NOAA's solar calculator for precise positions.
- Northern latitudes are positive; southern latitudes are negative.
Solar declination: frequently asked questions
What is solar declination?
Solar declination is the angle between the Sun's rays and the plane of Earth's equator on a given day. It ranges from about plus 23.44 degrees at the June solstice to minus 23.44 degrees at the December solstice, and is zero at the equinoxes. It drives the changing length of day and the Sun's noon height through the year.
What formula does this use?
It uses the Cooper approximation: declination equals 23.45 degrees times the sine of 360 times (284 plus the day number) divided by 365. This is a widely used engineering approximation accurate to within about a degree, suitable for solar geometry estimates.
What is the day number?
The day number N is the day of the year, with 1 being January 1 and 365 (or 366 in a leap year) being December 31. For example, the summer solstice around June 21 is day number 172. Enter the day number for your date.
How accurate is the Cooper approximation?
It is accurate to roughly one degree across the year, which is fine for solar panel tilt planning, daylight estimates, and educational use. For precise astronomical work use a full ephemeris from the U.S. Naval Observatory or NOAA's solar calculator.
Why does declination matter for solar energy?
The Sun's declination, combined with your latitude, sets the Sun's elevation at solar noon. That elevation determines the optimal tilt of fixed solar panels and the seasonal variation in available sunlight, which is central to solar energy design.
Official sources
- NOAA Global Monitoring Laboratory: solar position calculator.
- U.S. Naval Observatory: Sun position and declination data.
Reviewed by the CalculatorHub team, edited by James Graham, 17 June 2026. See our methodology.