Solar Declination Calculator

Solar declination (δ) is the Sun’s angular position north or south of Earth’s equatorial plane.

Cooper approximation

δ = 23.45° · sin( 360° · (284 + n) / 365 )

Series method (NOAA / Spencer style)

γ = (2π/365) · (n − 1 + (hour − 12)/24)

δ = 0.006918 − 0.399912 cosγ + 0.070257 sinγ − 0.006758 cos2γ + 0.000907 sin2γ − 0.002697 cos3γ + 0.00148 sin3γ

In the series method above, δ is in radians. This calculator converts it to degrees for display.

Optional estimates with latitude

• Solar noon altitude: h ≈ 90° − |φ − δ|
• Day length: cos ω_s = −tan φ tan δ, then day length ≈ (2/15)·ω_s (hours)

These are standard quick approximations and ignore refraction and terrain.

1) What solar declination means

Solar declination (δ) is the angle between the Sun’s rays and Earth’s equatorial plane. It tells you how far north or south the Sun appears at true solar noon, and it drives the seasonal shift of daylight, sunrise direction, and noon height.

2) Why declination changes through the year

Earth’s axis is tilted about 23.44° relative to its orbital plane. As Earth orbits the Sun, the tilt direction stays nearly fixed in space, so the Sun’s apparent position swings from south to north and back again. The result is a smooth yearly curve for δ.

3) Typical declination range and sign

Declination stays within roughly −23.44° to +23.44°. Positive values mean the Sun is north of the equator, which generally favors longer days in northern latitudes. Negative values favor longer days in southern latitudes. Within the tropics (|latitude| < 23.44°), δ can match latitude and the Sun can pass overhead.

4) Key dates: equinoxes and solstices

Near the March and September equinoxes, δ is close to 0°, so many locations see near‑equal day and night. Around June 21, δ approaches +23.44° (northern solstice). Around December 21, δ approaches −23.44° (southern solstice). The exact peak value varies slightly by year and model.

5) Day-of-year input and leap years

This calculator can work from a calendar date or from day-of-year (n). For day-of-year, choose the correct year so n runs 1–365 or 1–366 in leap years. That one‑day offset can shift δ by a few tenths of a degree near steep parts of the curve, which matters for tight planning.

6) Method choices and accuracy ideas

Three methods are offered. The Cooper sine approximation is fast and widely used for quick checks. The Spencer and NOAA-style series methods model the yearly curve more closely and typically reduce error, especially away from mid‑season. The NOAA option also accepts a solar-time hour so you can include a small intraday adjustment when needed.

7) Add latitude to get practical outputs

When you enter latitude (φ), the calculator estimates solar-noon altitude using h ≈ 90° − |φ − δ|. For example, at φ = 40° and δ = +23.44°, noon altitude is about 73.44°. It also estimates day length using cos ωs = −tan φ tan δ and day length ≈ (2/15)·ωs. At high latitudes, the model can reach 24 hours (midnight sun) or 0 hours (polar night).

8) Common uses in planning and analysis

Declination supports sunrise and sunset approximations, solar panel tilt studies, outdoor heat risk planning, and astronomy sessions. The same δ that raises the summer noon Sun also increases potential daily solar exposure at mid‑latitudes. It is also useful for agriculture calendars, building shading checks, and comparing seasonal angles across different cities.

What is a normal solar declination value?

Solar declination stays between about −23.44° and +23.44°. Values near 0° occur around equinoxes, while values near the extremes occur near the solstices.

Why do different methods give slightly different results?

Each method is an approximation. Simple sine formulas trade speed for accuracy, while series methods better match the annual curve. Small differences of tenths of a degree are common.

Do I need latitude to compute declination?

No. Declination depends mainly on the day of the year. Latitude is optional and is only used here to estimate solar-noon altitude and approximate day length.

What does a positive declination mean?

A positive declination means the Sun is north of the equator. Northern latitudes usually get higher noon Sun and longer daylight, while southern latitudes get the opposite.

How accurate is the day length estimate?

It is a quick geometric estimate based on sunrise hour angle. It ignores refraction, elevation, and local terrain, so real sunrise and sunset times can differ by several minutes.

Why is there a solar time hour input?

The NOAA-style series can include a small intraday adjustment using solar time. Keeping it at 12 works well for most purposes, but other values can refine the series output.

Can the Sun be directly overhead?

Yes, but only within the tropics. When declination is close to your latitude (and |latitude| is under about 23.44°), the Sun can pass near the zenith at solar noon.