How this instrument works
The solar elevation angle α measures how far the sun sits above a flat, open horizon — 0° at sunrise or sunset, climbing toward its daily peak at solar noon. The formula sinα = sinφ·sinδ + cosφ·cosδ·cosH comes from the spherical law of cosines applied to the astronomical triangle: the celestial pole, the zenith, and the sun, treated as three vertices on the surface of a sphere rather than a flat plane, because latitude and declination are themselves spherical coordinates and ordinary plane trigonometry cannot connect them.
Latitude φ fixes where you stand on Earth. Solar declination δ — the sun's angular distance north or south of the celestial equator — swings between about +23.45° and −23.45° across the year as Earth's tilted axis leans toward and away from the sun. Hour angle H measures how far the sun has moved from local solar noon, advancing at a steady 15° per hour, negative before noon and positive after. Set H to zero and the cosine term reaches its maximum of 1, which is exactly why solar noon gives the day's highest elevation for whatever declination is current.
The formula assumes a flat, unobstructed horizon and pure geometry — it ignores atmospheric refraction, which bends low sunlight upward by roughly half a degree near the horizon, so the sun stays visible for a few minutes after this instrument would report 0° or below. It also assumes the hour angle already accounts for the gap between clock noon and true solar noon, a gap that shifts with longitude inside a time zone and with the equation of time, which alone can move true noon up to about sixteen minutes from clock noon over a year.
- Enter Latitude in degrees, positive north of the equator and negative south — 40 for 40°N, −33 for a southern-hemisphere city near 33°S.
- Enter Solar declination in degrees; it runs from about −23.45° in December to +23.45° in June and crosses 0° at the spring and autumn equinoxes.
- Enter Hour angle from solar noon in degrees — 0° at true solar noon, then 15° for each hour away from it, negative before noon and positive after.
- Check sin(elevation), the intermediate sine value the spherical-triangle formula produces before the final angle is extracted.
- Read Solar elevation angle, the arcsine of that value, for the sun's height above the horizon in degrees.
Worked example — 40°N at equinox solar noon
Stand at 40°N latitude — Beijing, Madrid, and Philadelphia all sit close to this line — on the March or September equinox, when solar declination is 0°, and check the sun's height right at solar noon, hour angle 0°. Enter lat = 40, decl = 0, hourAngle = 0. The instrument computes sinα = sin(40°)·sin(0°) + cos(40°)·cos(0°)·cos(0°), which collapses to cos(40°) = 0.766044443119.
Taking the arcsine gives α = 0.872664625997 radians, which the instrument reports as 50.000000° of elevation. That figure is exactly 90° minus the 40° latitude — a shortcut that holds only at the two equinoxes, when the sun sits directly over the equator and every latitude's noon elevation reduces to its own complement.
Questions
Why is noon elevation 90° minus latitude only at the equinoxes?
Because that shortcut assumes declination is exactly 0°, true only on the two equinoxes. At solar noon (H = 0°) the formula reduces to sinα = sinφ·sinδ + cosφ·cosδ. With δ = 0° that simplifies to sinα = cosφ, so α = 90° − φ. On any other date δ is nonzero and the sun sits higher, near the summer solstice, or lower, near the winter solstice, than that simple complement.
What happens to noon elevation when declination is nonzero?
It shifts the 90° − φ result by roughly the declination itself. At 40°N on the summer solstice, δ near 23.45°, noon elevation rises to about 73.45°; on the winter solstice, δ near −23.45°, it falls to about 26.55°. That seasonal swing, not distance from the sun, is why winter sunlight arrives weaker and lower even at the same clock hour.
Can the elevation angle come out negative?
Yes. A negative α means the sun is below the horizon, and arcsine handles it without complaint since it is defined for any input from −1 to 1. Large hour angles paired with a low or negative declination push sinα below zero before sunrise or after sunset. The instrument does not clip the result, so a negative reading is real information, not an error state.
How do I convert a clock time into an hour angle?
Multiply the gap between local time and true solar noon by 15° per hour: H = 15° times the hours from solar noon. True solar noon is rarely exactly 12:00 — it drifts with your position inside your time zone and with the equation of time, which alone can shift clock noon up to about sixteen minutes from true solar noon over the course of a year.
Does this formula include atmospheric refraction?
No — it returns the sun's true geometric elevation, as if light travelled in perfectly straight lines. Earth's atmosphere actually bends low-angle sunlight upward by around 0.5° near the horizon, so the sun stays visible for a few extra minutes after this formula reports 0° or below, a correction surveyors and photographers apply separately.
Who actually uses a solar elevation formula like this one?
Solar installers use it to size panel tilt and predict shading through the day; architects run it across a year of hour angles to check whether a roof overhang blocks high summer sun while still admitting low winter sun; photographers use it to time golden hour, which starts once α drops below roughly 10°.