Calculator
Atmospheric Refraction
Earth's atmosphere bends light passing through it. This bending always deflects light upward. As a result, a celestial body appears slightly higher than its true position. This difference is called atmospheric refraction.
Refraction is strongly dependent on altitude. At the horizon (0° altitude) it is approximately 34 arcmin (slightly more than half a degree), at 45° altitude approximately 1 arcmin, and at the zenith it is zero. This is why when the Sun or Moon appears to touch the horizon, it is actually geometrically below the horizon. The "contact" we see is entirely due to refraction.
h: true altitude (degrees). Temperature and pressure correction is applied separately.
Refraction is critically important in heliacal rising calculations. Even when a star or planet is geometrically 1° below the horizon, it can be observed thanks to refraction. What Ptolemy and later authors called "first appearance at the horizon" is actually the refraction-corrected apparent altitude.
| True Altitude | Refraction | Apparent Altitude | Note |
|---|---|---|---|
| -0.57° | ~34.5' | 0° | Sun's upper limb at the horizon (moment of rising/setting) |
| 0° | ~34' | ~0.57° | Geometric horizon |
| 5° | ~9.9' | ~5.16° | Heliacal rising observation zone |
| 10° | ~5.3' | ~10.09° | |
| 30° | ~1.7' | ~30.03° | Effect drops to negligible levels |
| 90° | 0' | 90° | Zenith: refraction is zero |
Topocentric Parallax
Celestial body positions are typically calculated from the center of the Earth (geocentric). However, the observer is on Earth's surface. This positional difference creates a measurable angular shift, especially for nearby objects. This is called topocentric parallax.
Unlike refraction, parallax shifts the body downward (i.e., it lowers the altitude). The effect increases as the body approaches the horizon (maximum at the horizon, zero at the zenith) and is inversely proportional to the body's distance.
For the Moon HP ≈ 57', for the Sun HP ≈ 8.8" (0.147'), for planets and stars it is negligible.
The Moon is the closest celestial body to Earth (average ~384,400 km). Therefore its parallax (~57 arcmin, nearly 1°) is very significant. The Sun's parallax is only ~8.8 arcseconds, which is negligible for most applications. For planets and stars, parallax is effectively zero. In practice, topocentric parallax essentially means "lunar parallax."
| Celestial Body | Mean Distance | Horizontal Parallax | Effect Level |
|---|---|---|---|
| Moon | 384,400 km | 57' 02" | Very large, must always be calculated |
| Sun | 149,600,000 km | 8.8" | Small, relevant for precision work |
| Venus (inf. conj.) | ~41,000,000 km | ~32" | Generally negligible |
| Mars (conjunction) | ~78,000,000 km | ~18" | Negligible |
| Fixed Star | > 4 light-years | ~0" | Zero |
Combined Effect
The apparent altitude is found by applying both refraction and parallax together. The two effects work in opposite directions:
For stars P = 0, so h' = h + R. For the Moon, P can be close to or greater than R.
Consider the Moon at the horizon: refraction pushes it ~34' upward, but parallax pulls it ~57' downward. The net effect is that the Moon appears approximately 23' lower than its true position. For stars and planets, it is the exact opposite: they always appear higher.
When the Moon rises above the horizon, it must be geometrically about 23' above the horizon, because parallax exceeds refraction. In other words, unlike sunrise, moonrise does not occur when the Moon truly reaches the horizon, but slightly after.
Practical Use Scenarios
Astrophotography
When photographing celestial bodies near the horizon, refraction directly affects the image. The effect is not limited to positional shift: the vertical flattening of the Sun or Moon at the horizon is also a result of refraction. The bottom edge of the body is refracted more than the top edge (because it is closer to the horizon), making the disk appear elliptical.
At the horizon, refraction at the bottom edge of the disk (~34') is about 6' more than at the top edge (~28'). This difference compresses the visible disk's vertical diameter by approximately 19%. This is the reason for the "oval Sun" effect in photographs. At higher altitudes (10° and above), the refraction difference across the disk diameter drops to negligible levels.
Golden Hour and Sun Position
"Golden hour" calculations are based on the geometric Sun position. However, the Sun the observer sees is approximately 0.5° above its geometric position (when near the horizon) due to refraction. This means the Sun remains visible for several minutes even after its geometric sunset. Click the "Golden Hour Sun" preset in the calculator above to see the refraction effect at 3° altitude.
Moon Composition and Positional Accuracy
When planning a photograph of the Moon rising behind a building or mountain, the Moon's position is critically important. When the Moon is at 10° altitude, refraction pushes it ~5' upward, but parallax pulls it ~56' downward. As a net result, the Moon appears about 51' (nearly 1°) lower than the position you calculated. At focal lengths of 200mm and above, this difference directly affects composition.
Effect of Temperature Difference
A celestial body at the same altitude experiences different amounts of refraction on a hot summer night (35°C) versus a cold winter night (-15°C). Cold air means a denser atmosphere, which bends light more. Compare the "Moon (Hot Night)" and "Moon (Winter Night)" presets to see this difference. At 5° altitude, the difference can reach approximately 2 arcmin (118 arcseconds). This creates a difference that cannot be ignored in telescopic photography.
High Altitude Observation
When observing from a mountain at 3000 meters, atmospheric pressure drops to approximately 70% of sea level. This noticeably reduces refraction. A star near the horizon appears ~34' higher at sea level, but only ~24' higher at 3000 meters. Test this effect with the "Mountain Obs." preset.
Celestial Navigation
When sailors calculate positions from star and Sun observations, they apply refraction corrections to sextant measurements. Without correction, position errors near the horizon can reach up to 30 nautical miles (56 km). When the Moon is used, parallax correction is also mandatory.