Solar Declination Angle (δ), Sunrise Hour Angle & Daylight Duration Hours Calculator

Calculate solar declination δ, solar noon sun altitude (°), and daylight duration hours for any latitude.

Astronomical Daylight Duration
14.921 hours
Solar Declination Angle (δ) 23.45 degrees
Solar Noon Peak Sun Altitude Angle 72.75 degrees
Sunset Hour Angle (ω_s) 111.907 degrees
Inputs
⚡ Scenarios:
° lat
Observer latitude φ in degrees (−66° South to +66° North).
day N
Ordinal day of the year N (80 = Vernal Equinox, 172 = Summer Solstice, 266 = Autumnal Equinox, 355 = Winter Solstice).
Results & Graph
Astronomical Daylight Duration
14.921 hours
Solar Declination Angle (δ)
23.45 degrees
Solar Noon Peak Sun Altitude Angle
72.75 degrees
Sunset Hour Angle (ω_s)
111.907 degrees

Astronomical Daylight Duration

Graph 1 • Daylight Duration & Declination • Domain Architecture & Live Telemetry Maps Observer Latitude Degrees, Ordinal Day Of Year N through Daylight Duration & Declination Formula.
Solar Declination Angle (δ), Sunrise Hour Angle & Daylight Duration Hours Calculator • Custom Domain Schematic (Astronomical & Celestial Time) T_1T_2T_3T_4Chronological Span & Cadence (Daylight Duration & De) Observer Latitude Degrees: 40.70 ° lat • Ordinal Day Of Year N: 172 day N ASTRONOMICAL DAYLIGHT DURA 14.921 hours • Solar Declination Angl: 23.45 degrees • Solar Noon Peak Sun Al: 72.75 degrees Eq: δ = 23.45° × sin(360° × (284 + N
Graph 2 • Astronomical Daylight Duration Across Observer Latitude Degrees Sweep (40% → 160%) Evaluates δ = 23.45° × sin(360° × (284 + N) / 365); cos(ω_s) = −tan(φ) × tan(δ); across 7 operating points (current input highlighted).
0 hours 11.68 hours 23.36 hours 12.97 16.28° lat 13.51 24.42° lat 14.14 32.56° lat 14.92 40.70° lat 15.97 48.84° lat 17.58 56.98° lat 21.24 65.12° lat
Graph 3 • Joint Sensitivity: Observer Latitude Degrees × Ordinal Day Of Year N Compares Astronomical Daylight Duration when Ordinal Day Of Year N is shifted +25% (solid) vs. −25% (dashed).
0 hours 8.39 hours 16.79 hours --- Ordinal Day Of Year N @ 75% 13.30 -30% 13.66 -15% 14.08 +0% 14.60 +15% 15.26 +30%
Summary: At latitude 40.7° on Day 172, solar declination is δ = 23.45°, peak solar noon sun altitude is 72.7°, and astronomical daylight lasts 14.92 hours.
Summary: At latitude 40.7° on Day 172, solar declination is δ = 23.45°, peak solar noon sun altitude is 72.7°, and astronomical daylight lasts 14.92 hours.
⚡ Send result to: Unit Converter →Scientific →Date Difference →
Interactive chart & math engine powered by CalculatorGraph

Formula & Mathematical Methodology: Daylight Duration & Declination Formula

δ = 23.45° × sin(360° × (284 + N) / 365)
cos(ωs) = −tan(φ) × tan(δ)
DaylightHours = (2 / 15) × ωsdegrees
δ = 23.45° × sin(360° × (284 + N) / 365); cos(ω_s) = −tan(φ) × tan(δ); DaylightHours = (2 / 15) × ω_s_degrees
  • observerLatitudeDegrees: Observer latitude φ in degrees (−66° South to +66° North).
  • ordinalDayOfYearN: Ordinal day of the year N (80 = Vernal Equinox, 172 = Summer Solstice, 266 = Autumnal Equinox, 355 = Winter Solstice).

Rounding: Internal floating-point calculations run in 64-bit IEEE-754 double precision; display values round to 2–4 decimal places.

Worked Example: Daylight Duration & Declination Worked Example

Calculate solar declination δ, solar noon sun altitude (°), and daylight duration hours for any latitude.

  1. Enter the input measurement, system configuration, or astronomical parameter.
  2. Apply the standard conversion or analytical equation: δ = 23.45° × sin(360° × (284 + N) / 365); cos(ω_s) = −tan(φ) × tan(δ); DaylightHours = (2 / 15) × ω_s_degrees.
  3. Inspect the primary metric, secondary equivalents, and 3 real-time SVG graphs.
Result: Computes Daylight Duration & Declination via δ = 23.45° × sin(360° × (284 + N) / 365); cos(ω_s) = −tan(φ) × tan(δ); DaylightHours = (2 / 15) × ω_s_degrees.

Frequently Asked Questions

How does the Solar Declination Angle (δ), Sunrise Hour Angle & Daylight Duration Hours Calculator perform the calculation?

It evaluates δ = 23.45° × sin(360° × (284 + N) / 365); cos(ω_s) = −tan(φ) × tan(δ); DaylightHours = (2 / 15) × ω_s_degrees directly in your browser and renders three custom SVG charts in real time.

Can I solve backwards for an input parameter using Goal Seek?

Yes. Click "🎯 Goal Seek" in the results toolbar to solve backwards for any numeric input field.

Does this tool work offline?

Yes. All calculations run 100% locally in your browser with full offline PWA support.

Dates & Time · Astronomical & Celestial Time

Solar Declination Angle (δ), Sunrise Hour Angle & Daylight Duration Hours Calculator: inputs, method, and how to read the result

Calculate solar declination δ, solar noon sun altitude (°), and daylight duration hours for any latitude.

Choose inputs for the case you are checking

The defaults are illustrative. Replace them with values for your case and keep the units shown beside each field.

  • Observer Latitude Degrees: Observer latitude φ in degrees (−66° South to +66° North).
  • Ordinal Day Of Year N: Ordinal day of the year N (80 = Vernal Equinox, 172 = Summer Solstice, 266 = Autumnal Equinox, 355 = Winter Solstice).

What the calculation includes

With the default values shown, Astronomical Daylight Duration is 14.921 hours. The calculation follows Daylight Duration & Declination Formula; the formula and variable definitions above show how the inputs combine.

One-input check: With the other defaults held constant, changing Observer Latitude Degrees from 40.7 ° lat to 44.77 ° lat changes Astronomical Daylight Duration from 14.921 hours to 15.398 hours. This is a comparison inside the calculator's stated model, not a prediction.

Where to use caution

  • Uses official NIST Special Publication 811, IEEE/RFC systems engineering, APA psychometric, and USNO astronomical formulas.
  • Empirical metallurgy and calendar conversions represent standard reference-condition approximations.

Scope: Global NIST, IEEE, APA & Astronomical Standard Check the assumptions and limitations against your situation before relying on the estimate.

More in Dates & Time: Unit Conversion Calculator, Scientific Calculator, Date Difference & Business Days Calculator.