The parallactic angle q can be computed from latitude (φ), declination (δ), and hour angle (H):
If you use right ascension (RA) and local sidereal time (LST), the hour angle is: H = LST − RA (in hours), then convert to degrees with H° = 15·H.
- Select a workflow that matches your data source.
- Enter observer latitude and target declination in degrees.
- Provide hour angle, or enter RA with sidereal time inputs.
- For UTC + longitude mode, enter UTC time and longitude.
- Click Calculate. The result appears above the form.
- Use the download buttons to save CSV or PDF.
| Latitude (°) | Declination (°) | Hour angle (h) | Hour angle (°) | Parallactic angle q (°) |
|---|---|---|---|---|
| 31.50 | 20.00 | 2.00 | 30.00 | 60.782043 |
| 45.00 | -10.00 | -3.00 | -45.00 | -32.554814 |
| -23.50 | -45.00 | 5.00 | 75.00 | 97.341322 |
Parallactic angle is a small concept with big impact in practical observing. It links your site geometry to how a target appears to rotate in the field of view. Use this page to compute q from hour angle, or from right ascension and sidereal time, then export the results for planning, imaging logs, or instrument setup.
What the parallactic angle represents
Parallactic angle q is the position angle between the great circle to the zenith and the hour circle through the target. In practice it tells you how the sky is rotated relative to your local vertical at that moment.
Why observers care about q
Knowing q helps you predict field rotation, keep diffraction spikes aligned, and set slit orientation for spectroscopy. Near the meridian and at high declination, rotation can change quickly, so a fresh q value improves framing, guiding, and derotator setup.
Inputs used by this calculator
This tool uses observer latitude φ, target declination δ, and hour angle H. If you do not know H, you can compute it from H = LST − RA. Hours convert to degrees with H° = 15 × H, so 1 minute equals 0.25°.
The core trigonometric model
The calculation uses a stable quadrant‑aware atan2 form: q = atan2(sinH, tanφ·cosδ − sinδ·cosH). The atan2 output naturally ranges from −180° to +180°, which is convenient for plotting, plate solving overlays, and mount comparisons.
Interpreting the sign
Different software may swap sign conventions or reference axes. A positive q here follows the atan2(sinH, …) definition with north and east on the sky. If your mount reports the opposite sign, multiply by −1 to match it consistently.
When values become sensitive
q can be sensitive when the denominator term approaches zero, which happens near the zenith for some geometries. Small errors in time, longitude, or RA then shift H, changing q noticeably. Using UTC + longitude mode and accurate coordinates reduces drift.
Practical imaging guidance
For long exposures on an alt‑az mount, increasing rotation demands shorter sub‑frames or a field derotator. If q changes by several degrees during a run, stars may smear at the edges. Re‑compute q every 15–30 minutes as hour angle evolves.
Checking results with example data
Try latitude 31.5°, declination 20°, and H = 2 h (30°). The sample table shows q near 60.78°. Changing H to −3 h (−45°) flips the geometry and can produce negative q, matching the expected east‑of‑meridian behavior. Use this as a quick sanity check.
1. What is the parallactic angle used for?
It is used to describe sky orientation relative to the local vertical. Observers use it to anticipate field rotation, align a spectrograph slit, and match camera framing between sessions.
2. Do I need right ascension to compute it?
No. If you already know hour angle H, enter it directly. RA is only needed when you compute H from LST or from UTC plus longitude.
3. Why does the calculator offer three modes?
Different workflows provide different inputs. Some observers track hour angle, some have LST from software, and others only know UTC time and site longitude. All modes feed the same final formula.
4. What range should q fall in?
This implementation returns q in degrees from −180° to +180°. You can unwrap it for plots, or add or subtract 360° to match a preferred continuous range.
5. How accurate is the UTC + longitude sidereal time step?
It uses a standard approximate GMST model from Julian date. It is typically adequate for planning and framing, but precision applications should use high‑accuracy ephemeris software and correct site coordinates.
6. Why does q change fast near the meridian?
As the target crosses the meridian, the hour angle changes sign and the geometry relative to the zenith can shift rapidly. That can make the atan2 inputs vary quickly, increasing the apparent rotation rate.
7. What if my mount shows the opposite sign?
Sign conventions differ. Keep one convention across your workflow. If needed, multiply the reported value by −1 to match your mount or software, and document that choice in your notes.