Calculator Inputs
Plotly Graph
This graph shows sun altitude across local solar time for the current latitude and declination.
Example Data Table
| Scenario | Latitude (°) | Day of Year | Solar Time (h) | Declination (°) | Hour Angle (°) | Sun Altitude (°) |
|---|---|---|---|---|---|---|
| Equator, equinox, morning | 0.00 | 80 | 9.00 | -0.40 | -45.00 | 45.00 |
| Mid-latitude, equinox, noon | 35.00 | 80 | 12.00 | -0.40 | 0.00 | 54.60 |
| Northern summer, noon | 51.50 | 172 | 12.00 | 23.45 | 0.00 | 61.95 |
| Northern winter, noon | 51.50 | 355 | 12.00 | -23.45 | 0.00 | 15.05 |
| Southern hemisphere, afternoon | -33.90 | 20 | 15.00 | -20.34 | 45.00 | 48.09 |
Formula Used
The calculator uses the standard solar altitude relation:
sin(α) = sin(φ) sin(δ) + cos(φ) cos(δ) cos(H)
- α = sun altitude angle
- φ = observer latitude
- δ = solar declination
- H = solar hour angle
The automatic mode also estimates declination from day number using:
δ ≈ 23.45 × sin[(360/365) × (284 + n)]
Here, n is the day of the year. Hour angle is found from local solar time:
H = 15 × (solar time − 12)
Zenith angle is then:
Zenith = 90 − altitude
How to Use This Calculator
- Select the calculation mode.
- Enter latitude in degrees. North is positive. South is negative.
- For automatic mode, enter day of year and local solar time.
- For custom mode, enter solar declination and hour angle directly.
- Choose your preferred decimal precision.
- Press Calculate Sun Altitude.
- Review the result card above the form.
- Use the CSV or PDF buttons to export your results.
- Check the graph to understand how altitude changes across the day.
FAQs
1) What is sun altitude?
Sun altitude is the angle between the Sun and the local horizon. A higher altitude means the Sun appears higher in the sky. At sunrise and sunset, the altitude is near zero. Negative altitude means the Sun is below the horizon.
2) What is the difference between altitude and zenith angle?
Altitude is measured upward from the horizon. Zenith angle is measured downward from the point directly overhead. They always add to 90 degrees. If altitude rises, zenith angle falls by the same amount.
3) Why does the calculator use local solar time?
Solar geometry formulas are naturally based on the Sun’s actual position relative to local noon. Clock time can differ from solar time because of longitude, time zones, and the equation of time.
4) Can the altitude become negative?
Yes. A negative result means the Sun is below the horizon. This commonly happens before sunrise, after sunset, or during polar night conditions at extreme latitudes.
5) Does this include atmospheric refraction?
No. The core altitude result is geometric. Atmospheric refraction slightly lifts the apparent Sun near the horizon, but that small correction is not added to the reported altitude here.
6) Why does daylight duration change with latitude?
Earth’s axis is tilted. As declination changes through the year, the Sun follows different paths across the sky. Higher latitudes experience larger seasonal swings in daylight duration than places near the equator.
7) Can I use southern hemisphere locations?
Yes. Enter southern latitudes as negative values. The formulas work for both hemispheres, and the calculator will still compute declination, altitude, zenith angle, and daylight duration correctly.
8) How accurate is the declination estimate from day number?
The declination formula used here is a common approximation and is suitable for most educational, engineering, and planning tasks. For high-precision astronomy, more advanced solar ephemeris models are preferred.