Calculator
Formula Used
The zenith angle z is the angle between the local vertical (zenith direction) and the line of sight to a target. For an altitude (elevation) angle a above the horizon:
z = 90° − a
For equatorial coordinates using observer latitude φ, target declination δ, and hour angle H:
cos(z) = sin(φ)sin(δ) + cos(φ)cos(δ)cos(H)
The time-based method computes local sidereal time (LST) from the observation UTC time and longitude, then uses H = LST − RA with the same cosine relation. An optional air mass estimate is provided for z < 90° using a standard approximation.
How to Use This Calculator
- Select a method that matches your available inputs.
- Enter angles in the selected units and keep sign conventions consistent.
- For the sidereal method, enter UTC time and longitude (East positive).
- Click Calculate to view results above the form.
- Use the CSV or PDF buttons to export the computed values.
Example Data Table
| Scenario | Inputs | Key Output |
|---|---|---|
| Altitude method | Altitude = 35.5° | Zenith ≈ 54.5° |
| Coordinates method | Latitude = 25°, Declination = 10°, Hour Angle = 30° | Zenith computed from cosine formula |
| Sidereal method | UTC time + Longitude + RA/Dec + Latitude | Hour angle and zenith derived automatically |
1. Zenith angle in one sentence
The zenith angle z is the separation between the local vertical and the target direction, so z = 0° is straight overhead and z = 90° is on the horizon. Because elevation a = 90° − z, both angles describe the same sky position.
2. Why observers care about z
Light travels through more atmosphere at larger zenith angles, increasing extinction, scattering, and seeing blur. A common planning rule is to keep important targets at z ≤ 60° (elevation ≥ 30°) to reduce losses. Near the horizon, refraction and local obstructions dominate the error budget.
3. Air mass numbers you can use
This calculator reports an approximate air mass for z < 90°. Typical values are about 1.00 at z=0°, 2.00 at z=60°, ~3.8 at z=75°, and ~5.6 at z=80°. These estimates help compare exposure time, transparency, and signal-to-noise planning.
4. Coordinate method: φ, δ, and H
If you know observer latitude φ, declination δ, and hour angle H, the cosine relation gives cos(z) directly. Hour angle changes at roughly 15° per hour, so a small timing error can shift z noticeably for fast-moving targets.
5. Time method: UTC, GMST, and LST
The time-based option computes GMST from the Julian Date, then adds longitude to obtain LST. With H = LST − RA, the same cosine formula yields z. Remember the sidereal day is about 23 h 56 m, so star times drift versus clock time each night.
6. Solar geometry and engineering uses
For sunlight, zenith angle controls the direct-beam projection through cos(z). At z=60°, the projected intensity is roughly half of overhead conditions, ignoring atmospheric losses. Solar PV, thermal collectors, and daylight studies often combine z with azimuth to estimate incident irradiance.
7. Practical limits and edge cases
When a target is below the horizon, elevation becomes negative and z exceeds 90°. Air mass is not meaningful there, so the calculator shows it as N/A. Close to z≈90°, small rounding or refraction differences can change elevation by noticeable arcminutes.
8. Tips for cleaner results
Use consistent sign conventions: north latitudes positive, east longitudes positive, and declination north positive. If your RA is in hours, keep it in hours and let the calculator convert using 15° per hour. For reproducibility, record the exact UTC timestamp used in the sidereal method.
FAQs
1) What is the difference between zenith angle and elevation?
Elevation is measured up from the horizon, while zenith angle is measured down from the vertical. They are complementary: z = 90° − a. Use whichever matches your instrument or planning chart.
2) Can zenith angle be greater than 90°?
Yes. If the target is below the horizon, elevation is negative and the zenith angle exceeds 90°. In that case, air mass is not physically useful, so the calculator reports it as N/A.
3) Why does the calculator ask for UTC time in the sidereal method?
UTC avoids timezone offsets and daylight-saving shifts. Sidereal time is derived from Earth rotation and Julian Date, so a clean UTC timestamp reduces common input mistakes and makes results easier to reproduce.
4) What longitude sign should I use?
Enter longitude as positive east of Greenwich and negative west. If your map lists “W”, use a negative value. If it lists “E”, use a positive value. This keeps LST and hour angle consistent.
5) How accurate is the air mass value?
It is an approximation intended for planning. It is reliable for typical observing zenith angles and degrades near the horizon where refraction, terrain, and local weather dominate. Use measured extinction for precision work.
6) Can I use this for satellites or fast movers?
Yes, but time accuracy matters. Hour angle changes quickly, so use the time method with the correct UTC timestamp and updated RA/Dec or topocentric coordinates. For very fast passes, recompute frequently.
7) Why is cos(z) included in the results?
Cos(z) appears in projection and illumination calculations, including solar irradiance and some radiative-transfer models. It is also a convenient intermediate value for comparing elevations without converting back and forth.