Atmospheric Refraction

The Sun you see at the horizon is not where it actually is. Earth's atmosphere bends light, lifting the apparent Sun position by about 0.57° at the horizon. This is large enough to matter for prayer times.

What is refraction

Light travels in a straight line through a uniform medium. But the atmosphere is not uniform — it gets progressively thinner (less dense) with altitude. As sunlight enters the atmosphere from space, it passes through layers of increasing density, bending it toward the denser layer below.

The result: light from the Sun arrives at your eye on a curved path. The apparent direction of the Sun is higher than its geometric direction.

At the zenith (Sun directly overhead), refraction is negligible — the light arrives nearly perpendicular to the atmospheric layers. Near the horizon, the light travels through the longest atmospheric path, and refraction is largest: approximately 0.57° for standard sea-level conditions.

This is why the Sun appears to still be above the horizon at geometric sunset — its light is bent around the curve of the Earth, and you're seeing it through a thick atmospheric lens.

Standard refraction correction

For sunrise and sunset calculations, the standard correction combines:

Atmospheric refraction at the horizon: ~0.567°
Sun's semi-diameter: ~0.267° (the Sun's disk is ~0.533° wide; sunrise is when the upper limb, not center, crosses the horizon)

Combined correction: ~0.8333° (50 arcminutes)

This means geometric sunrise is defined as the moment when the Sun's center is at −0.8333° altitude — accounting for the disk and refraction together.

For Fajr and Isha, the target altitude is −(depression angle), e.g., −15° for ISNA Fajr. Refraction at these angles is small (about 0.02°) and is already included in the NREL SPA computation.

Bennett formula

The most widely used refraction approximation for altitudes above −5° is Bennett's formula (1982):

R = 1.02 / tan(h + 10.3 / (h + 5.11))

Where:

R is the refraction in arcminutes
h is the apparent altitude in degrees

At h = 0° (horizon): R ≈ 34.1 arcminutes (0.57°) At h = 5°: R ≈ 9.9 arcminutes At h = 10°: R ≈ 5.6 arcminutes At h = 45°: R ≈ 1.0 arcminute At h = 90°: R = 0 (no refraction at zenith)

Bennett's formula is accurate to about 0.07 arcminutes for standard atmospheric conditions. It is used in many prayer time libraries and almanac software.

Pressure and temperature effects

Refraction depends on air density, which varies with pressure and temperature. The NREL SPA applies a correction factor:

R_corrected = R × (P / 1010) × (283 / (273 + T))

Where:

P is surface pressure in millibars (standard: 1013.25 mb)
T is surface temperature in °C (standard: 10°C)

Altitude effects: At 3,000 metres elevation (e.g., cities in the Ethiopian highlands or mountain cities in Pakistan), pressure is about 710 mb — 70% of sea level. Refraction is reduced proportionally, shifting sunrise/sunset by about 20–25 seconds.

Temperature effects: A cold winter day at −20°C vs. a hot summer day at 40°C changes refraction by about 6%, shifting times by 10–15 seconds.

For most prayer time applications, the standard correction is sufficient. The nrel-spa package exposes pressure and temperature parameters for those who need exact corrections for high-altitude or extreme-climate locations.

Implications for prayer times

Refraction primarily affects:

Sunrise/Shuruq — the 0.8333° correction is essential; without it, computed sunrise would be ~2 minutes late
Maghrib — same correction applies at sunset
Fajr/Isha — small refraction at −15° to −20° altitude (< 0.02°), negligible in practice
Extreme altitude locations — Mecca (~270m), Medina (~620m), Kabul (~1,800m), Addis Ababa (~2,300m) all have noticeably reduced refraction

The pray-calc package uses the full NREL SPA refraction model including pressure and temperature corrections when those inputs are provided.