Calculator
Formula used
Solar declination angle describes the sun’s angular position north or south of the equatorial plane. A widely used accurate approximation is the Spencer expression:
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)
Γ = 2π(n − 1) / 365
Here, δ is declination in radians and n is the day number. This page also supports the Cooper and simple sine forms for comparison.
How to use this calculator
- Select whether you want to enter a calendar date or day number.
- Choose the preferred declination formula.
- Enter the year to handle leap-year day counts correctly.
- Set chart window days to control the local trend plot range.
- Choose decimal precision for output values.
- Press the calculate button to display the result above the form.
- Review the charts, interpretation notes, and annual pattern.
- Use the CSV or PDF buttons to export the displayed result.
Example data table
| Date | Day Number | Approx. Declination | Season Marker |
|---|---|---|---|
| 2026-03-20 | 79 | ~ 0.00° | March equinox |
| 2026-06-21 | 172 | ~ +23.44° | June solstice |
| 2026-09-22 | 265 | ~ 0.00° | September equinox |
| 2026-12-21 | 355 | ~ -23.44° | December solstice |
Why this calculator is useful
- Helps estimate sun position trends for solar design studies.
- Supports academic work in solar geometry and radiation analysis.
- Compares multiple formulas in one page.
- Provides fast outputs in degrees and radians.
- Includes exports and two Plotly visualizations.
FAQs
1. What is solar declination angle?
Solar declination angle is the sun’s angular location relative to Earth’s equatorial plane. It changes throughout the year because Earth’s axis is tilted during its orbit around the sun.
2. Why does declination matter in solar work?
Declination affects solar altitude, day length, and incident radiation geometry. It is a core input for panel tilt studies, shading reviews, and educational sun-path calculations.
3. Which formula should I choose?
Spencer is commonly used for better accuracy. Cooper is also popular and simple. The simple sine form is helpful for learning and quick approximate checks.
4. What is the day number?
Day number is the sequential day in the year. January 1 is day 1, and December 31 is day 365 or 366 in leap years.
5. Why are the solstice values near ±23.44°?
Earth’s axial tilt is about 23.44°. Around the June and December solstices, solar declination reaches values close to that magnitude north or south of the equator.
6. Can I use this for panel sizing?
It helps with solar geometry, but panel sizing needs more inputs. Use declination along with latitude, irradiance, azimuth, tilt, and system-loss assumptions.
7. Why does the result appear above the form?
The layout is arranged so you see outputs immediately after submission. This makes result review, export actions, and chart interpretation easier on long pages.
8. Does the calculator support leap years?
Yes. The page checks the selected year and adjusts the allowed day-number range. That keeps date conversion and validation consistent during leap years.