Calculator Inputs
Plotly Graph
The chart shows how solar zenith, elevation, and azimuth vary across the selected date at your location.
Example Data Table
| Scenario | Latitude | Longitude | UTC Offset | Date | Time | Use Case |
|---|---|---|---|---|---|---|
| Karachi Noon Study | 24.8607 | 67.0011 | +5 | 2026-06-21 | 12:00 | Summer zenith comparison |
| Equatorial Reference | 0.0000 | 37.9062 | +3 | 2026-03-20 | 12:00 | Equinox alignment check |
| London Winter Check | 51.5074 | -0.1278 | 0 | 2026-12-21 | 10:30 | Low solar elevation review |
| Sydney Morning Review | -33.8688 | 151.2093 | +10 | 2026-01-15 | 09:15 | Azimuth tracking setup |
Formula Used
γ = (2π / daysInYear) × [(N − 1) + (hour − 12) / 24]
Et = 229.18 × [0.000075 + 0.001868 cos γ − 0.032077 sin γ − 0.014615 cos(2γ) − 0.040849 sin(2γ)]
δ = 0.006918 − 0.399912 cos γ + 0.070257 sin γ − 0.006758 cos(2γ) + 0.000907 sin(2γ) − 0.002697 cos(3γ) + 0.00148 sin(3γ)
Time Offset = Et + 4 × longitude − 60 × timezone
TST = local clock minutes + Time Offset
H = TST / 4 − 180
cos θz = sin φ sin δ + cos φ cos δ cos H
θz = arccos(cos θz)
Solar elevation = 90° − θz
Here, φ is latitude, δ is declination, H is hour angle, and θz is the solar zenith angle. The calculator also estimates azimuth, solar noon, sunrise, sunset, and air mass using the same solar geometry framework.
How to Use This Calculator
- Enter latitude in decimal degrees. North is positive.
- Enter longitude in decimal degrees. East is positive.
- Provide the local UTC offset used by the clock time.
- Select the local date and local time you want to analyze.
- Choose the graph interval in minutes for the daily curve.
- Press Calculate Solar Angles to show the results section above the form.
- Review zenith, elevation, azimuth, solar noon, and daylight metrics.
- Use the CSV or PDF buttons to export the calculated report.
Frequently Asked Questions
1) What is the solar zenith angle?
The solar zenith angle is the angle between the Sun and the vertical direction above a location. A smaller zenith angle means the Sun is higher overhead. When the angle reaches 90°, the Sun is on the horizon.
2) How is zenith angle related to solar elevation?
Solar elevation and solar zenith are complementary angles. Their sum is 90°. If the zenith angle is 25°, the elevation angle is 65°. This relationship makes both values easy to compare for energy, optics, and daylight studies.
3) Why do I need longitude and timezone?
Longitude and timezone are needed to convert clock time into true solar time. Two places can share the same civil time while the Sun is at different positions. These values correct the local clock to the Sun’s actual position.
4) What does the equation of time represent?
The equation of time measures the difference between apparent solar time and mean clock time. Earth’s orbital shape and axial tilt cause that difference. It slightly shifts solar noon and changes computed hour angle during the year.
5) Can this calculator help with solar panel planning?
Yes. Zenith and azimuth help estimate incoming sunlight direction, shading risk, and seasonal exposure patterns. This supports panel tilt studies, tracking analysis, and site comparisons. Final design still benefits from local weather and obstruction measurements.
6) Why can sunrise or sunset show as unavailable?
At very high latitudes, some dates have polar day or polar night. In those conditions, the Sun may stay above or below the horizon all day. The calculator flags this instead of reporting ordinary sunrise and sunset times.
7) What is relative air mass in this tool?
Relative air mass estimates how much atmosphere sunlight passes through compared with the vertical path at sea level. It grows as the Sun moves lower. Larger values often mean stronger scattering, higher attenuation, and weaker direct sunlight.
8) Is this calculator suitable for scientific work?
It is suitable for education, engineering estimates, quick planning, and many physics applications. It uses well-known solar position equations. For highest-precision observatory or legal work, compare results with an ephemeris or professional solar geometry library.