Solar Zenith Angle Calculator

Formula Used

The core solar zenith relationship is:
cos(θz) = sin(φ)·sin(δ) + cos(φ)·cos(δ)·cos(H)

• θz = solar zenith angle (degrees), measured from the vertical.
• φ = observer latitude (degrees).
• δ = solar declination (degrees), from Earth’s axial tilt and orbital position.
• H = hour angle (degrees), from true solar time.

This calculator derives δ and the equation of time using an advanced NOAA-style solar-position model, then computes true solar time, hour angle, and finally zenith, elevation, and azimuth. Atmospheric refraction is applied to provide corrected elevation and zenith.

How to Use This Calculator

1. Enter your local date and local time for the observation.
2. Set the timezone offset from UTC, and add DST minutes if needed.
3. Provide latitude and longitude in degrees (east and north positive).
4. Optionally enter pressure and temperature to refine refraction.
5. Click Calculate. Results appear above the form for easy review.
6. Use Download CSV or Download PDF to export your results.

Example Data Table

Solar Zenith Angle Overview

The solar zenith angle is the angle between incoming sunlight and the local vertical. At 0° the Sun is overhead; at 90° it sits on the horizon. Because many solar calculations scale with cos(θz), the zenith angle is a direct geometric driver of surface irradiance.

Why Zenith Angle Matters

As the Sun lowers, sunlight travels through more atmosphere, increasing absorption and scattering. This changes spectrum, visibility, and UV levels. Instruments such as pyranometers and UV sensors show strong diurnal signatures that closely track zenith angle, especially under clear-sky conditions.

Location and Time Inputs

Latitude and longitude define where you stand on Earth. Date and clock time determine Earth’s orbital position and rotation relative to the Sun. Timezone and daylight saving settings align civil time with local legal time; choosing the wrong offset can shift hour angle and solar noon noticeably.

Solar Geometry Outputs

The calculator estimates solar declination, equation of time, true solar time, and hour angle, then computes elevation and zenith angle. It also reports azimuth (compass direction) and can estimate sunrise, sunset, and day length when the Sun rises and sets at the chosen latitude and date.

Cosine Projection on a Flat Surface

For a horizontal surface, direct-beam contribution scales with cos(θz). At 60° zenith, the cosine factor is 0.5, so direct normal irradiance contributes half as much to a flat plane as it would with overhead Sun, before considering atmospheric losses and clouds.

Refraction Near the Horizon

Atmospheric refraction bends light downward, making the Sun appear slightly higher than its geometric position, most strongly near the horizon. Optional pressure and temperature inputs let the calculator apply a practical refraction correction, improving low-elevation angles and sunrise/sunset timing estimates.

Where Zenith Angle is Used

Solar energy studies use zenith angle for tracker control, panel tilt comparisons, and estimating cosine losses. Climate and hydrology models use it in radiation-driven terms like evapotranspiration. Remote sensing workflows apply it to interpret illumination geometry and normalize reflectance across seasons.

Tips for Accurate Use

Enter coordinates carefully; small longitude errors can shift timing results. Keep your time standard consistent when comparing with datasets (local time versus UTC). Document daylight saving choices for the selected date. For best agreement with observations, use measured pressure and temperature when refraction matters.

FAQs

1) What is the difference between solar elevation and solar zenith angle?

Solar elevation is the angle above the horizon. Solar zenith angle is measured from the vertical. They are complementary: elevation = 90° − zenith.

2) Why does my result change when I toggle daylight saving?

Daylight saving shifts civil clock time by one hour. That changes the computed true solar time and hour angle, so zenith and azimuth update even though the Sun’s physics is unchanged.

3) How accurate are sunrise and sunset times from this tool?

They are good for planning and education, typically within a few minutes for many locations. Accuracy depends on correct timezone, coordinates, and whether refraction is enabled, especially when the Sun is low.

4) What does “equation of time” mean?

It is the minutes difference between apparent solar time and mean clock time caused by Earth’s tilted axis and slightly elliptical orbit. It helps convert civil time to true solar time.

5) Can the Sun be above the horizon at midnight?

Yes, at high latitudes near summer the Sun may not set (midnight Sun). In that case, sunrise and sunset may be unavailable, but zenith and azimuth still compute for the selected time.

6) Why do I get “N/A” for sunrise or sunset?

At some latitudes and dates, the Sun may not rise or may not set. Numerical rounding can also affect borderline cases. Check the latitude, date, and whether it is polar day or polar night.

7) Which inputs matter most for zenith angle?

Latitude and true solar time dominate. Longitude and timezone affect the time conversion, while declination depends on date. Pressure and temperature mainly influence refraction when the Sun is near the horizon.