Solar Declination (Approximate) Calculator
Calculate the sun's declination angle for any day of the year — or find the days that produce a given declination — with step-by-step math, live seasonal charts, conversion tables and solar noon and day-length metrics.
Zero crossings at N ≈ 81 and 264
The (N − 81) offset places the equinoxes near 22 March and 21 September in the formula, when the sun sits over the equator (δ = 0).
Extremes near the solstices
δ peaks at +ε around N ≈ 172 (June) and bottoms at −ε around N ≈ 355 (December), where the seasons are most extreme.
Fastest change at the equinoxes
Declination shifts by roughly 0.4° per day near the equinoxes but barely moves near the solstices, which is why sunrise times shift quickly in spring and autumn.
Two days share each declination
Because the curve rises then falls, every δ between −ε and +ε occurs twice a year, once while rising and once while falling. The N solver reports both.
How to Use This Calculator
Declination links the day of the year N and the axial tilt ε to the sun's angle north or south of the equator. Latitude is optional and only feeds the site metrics.
1. Choose what to solve for
Declination from a day number, or the day number(s) that give a chosen declination.
2. Set the day
Use a preset, drag the slider, type N, or pick a calendar date — they all stay in sync.
3. Check the axial tilt
Leave it at 23.44° for Earth, or nudge it to 23.45° / 23.5° to match a textbook or reference.
4. Add a latitude (optional)
Enter your latitude to unlock noon altitude, day length and panel-tilt figures in the charts and Advanced mode.
5. Read results and charts
See δ in degrees, radians and DMS, then explore the declination, noon-altitude and day-length curves with your day marked.
6. Go Advanced
Advanced mode adds sensitivity, season phase, rate of change, sunrise/sunset in solar time and more.
What This Tool Offers
Built around the widely used sine approximation of solar declination, a staple of solar-energy and daylighting calculations.
Solve both directions
Find δ for a day, or find both days of the year that share a target δ.
Date picker sync
Choose a calendar date and the day number updates instantly, or the reverse.
Three live charts
Declination, solar-noon altitude and day-length curves with your day marked.
Monthly & key-date tables
Colour-graded declination by month, plus solstice and equinox values.
Angle converter
Convert declination between degrees, radians, gradians, arcminutes and DMS.
Site metrics
Noon altitude, day length, sunrise/sunset and a noon-optimal panel tilt for your latitude.
Conversion Tables & Reference
An angle converter plus live tables driven by your current axial tilt and latitude.
Angle converter
Declination by Month
Declination on the 1st of each month.
Solstices & Equinoxes (from the formula)
Days That Share a Given Declination
Each declination occurs twice a year.
What Is Solar Declination?
Solar declination is the angle between the sun's rays and the plane of Earth's equator. Because Earth's axis is tilted about 23.44° relative to its orbit, the declination swings between +23.44° at the June solstice and −23.44° at the December solstice, passing through 0° at the equinoxes. It determines the sun's midday height, the length of the day and the strength of the seasons at any latitude.
Approximate Declination Formula
Rearranged to find the day number from a declination (two solutions per year):
N = 81 + θ · 365/360
Variables
| Symbol | Meaning | Notes |
|---|---|---|
| δ | Solar declination | Degrees; positive when the sun is north of the equator |
| ε | Axial tilt (obliquity) | ≈23.44° for Earth |
| N | Day of the year | 1 January = 1 |
| 81 | Offset | Places δ = 0 near the March equinox |
| φ | Latitude (optional) | Used for altitude and day length |
Calculation Examples
θ = 360/365 × (172 − 81) = 89.75°
δ = 23.44 × sin(89.75°) = ≈ +23.44°
θ₁ = arcsin(10 / 23.44) = 25.25° → N₁ = 81 + 25.25 × 365/360 ≈ 106.6 (about 17 April)
θ₂ = 180° − 25.25° = 154.75° → N₂ ≈ 237.9 (about 26 August)
Answer: N ≈ 107 and N ≈ 238
Frequently Asked Questions
It is usually within about a degree of more rigorous formulas. It treats Earth's orbit as circular and uses a fixed 365-day year, so errors are largest around the equinoxes and in leap years.
The declination curve is a sine wave, so any value between −ε and +ε is reached once while it rises and once while it falls. Both are real dates in the same year.
At solar noon the sun's altitude is 90° − |φ − δ|. When δ equals your latitude, the sun passes directly overhead. The optional latitude field computes this along with day length.
Day 81 is around 22 March in a non-leap year, close to the March equinox, when the sun crosses the equator northward. Subtracting 81 makes the sine start at zero at that moment.
Yes, as an estimate. A panel tilted at |φ − δ| from horizontal faces the noon sun squarely. Year-round fixed tilts are usually set near the latitude, adjusted for seasonal priorities.
Beyond about 66.56° latitude the sun can stay above or below the horizon all day. The day-length metric shows 24 h or 0 h in those cases.
What it means
Solar declination δ is the angle between the Sun’s rays and the Earth’s equatorial plane. This approximation gives δ as a sinusoidal function of the day of the year (N_day), with amplitude 23.44° (obliquity of the ecliptic). It is used in solar energy calculations, in building design for passive solar heating, and in climatology. The formula is accurate to within a few degrees. More precise models include the Earth’s elliptical orbit. Understanding this helps in tracking the Sun’s position for various applications, including agriculture, architecture, and photovoltaic system design.
Worked example
Solar Declination – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| N_day | 81 |
| Parameter | Value |
|---|---|
| N_day | 172 |
Common mistakes
- Solar declination approximation: δ ≈ 23.44° × sin( (360/365)×(N_day − 81) ).
- N_day: Day of the year (1‑365) – January 1 = 1.
- Angles in degrees: Use degrees for both input and output – convert if using radians.
- Approximation: This is a sinusoidal approximation – more accurate models include orbital eccentricity.
- Leap year: For leap years, use 366 days – but this formula is often used with 365.
Applications
Solar declination (δ) is the angle between the Sun and the celestial equator, varying seasonally between +23.44° and −23.44°. The approximate formula δ ≈ 23.44° sin(360°/365 (N_day − 81)) gives the declination for a given day of the year. This is used in solar geometry calculations for solar energy, climatology, and navigation. Engineers designing solar tracking systems use it to compute the Sun's position. Meteorologists use declination to understand seasonal weather patterns. By calculating declination, one can determine solar altitude and azimuth, which are essential for shading analysis and daylighting design. This formula is a practical tool for a range of applications involving the Sun.
- Solar panel tilt and orientation optimisation
- Calculation of solar altitude and azimuth for shading studies
- Weather and climate modelling (insolation)
- Navigation and sundial design
- Education and demonstration of seasonal cycles
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