Atmospheric Refraction
When you watch a sunset, the Sun you see has already physically dipped below the horizon — you are looking at a ghost image, displaced upward by the bending of light through Earth's atmosphere. This phenomenon is called atmospheric refraction: the gradual curving of light rays as they pass through layers of air with continuously changing density. Because denser air has a higher refractive index, light slows and bends toward the denser medium, following a curved path rather than a straight line from source to observer. The practical consequences are significant and measurable. Stars appear slightly higher in the sky than their true geometric positions, celestial objects near the horizon are displaced most strongly (by up to about 0.5°, roughly one solar diameter), and the duration of daylight is extended by several minutes each day because we can see the Sun before it truly rises and after it truly sets. Surveyors, navigators, and astronomers must all apply refraction corrections to obtain accurate measurements. Atmospheric refraction also produces striking visual effects such as mirages, the green flash at sunset, and the flattened appearance of the Sun near the horizon. Understanding refraction requires combining Snell's Law of refraction with knowledge of how atmospheric density — and therefore refractive index — varies with altitude.
- Air Density and Refractive Index
- Applying Snell's Law Layer by Layer
- Zenith Angle Dependence
- Temperature and Pressure Corrections
- Mirages as Extreme Refraction
Air Density and Refractive Index
Air is not a single uniform medium. Its density decreases with altitude: air near the ground is compressed by the weight of all the air above it and is therefore denser. The refractive index n of air is very close to 1 but increases with density. At sea level, n ≈ 1.000293 for visible light; at high altitude, n approaches 1.000000 (vacuum). This small but real gradient in n means that as a light ray travels downward through the atmosphere, it continuously enters slightly denser air and bends toward the vertical — curving Earth-ward.
Applying Snell's Law Layer by Layer
Think of the atmosphere as thousands of thin horizontal layers, each with a slightly different refractive index. At each boundary, Snell's Law applies: n₁ sin θ₁ = n₂ sin θ₂. Because n₂ > n₁ (lower layer is denser), the refracted angle θ₂ is smaller, meaning the ray bends toward the normal (toward vertical). Summing these infinitesimal bends over the entire atmospheric path produces a net angular displacement that can be modelled mathematically. For a ray arriving at zenith angle z (angle from directly overhead), the total refraction R ≈ 1.02 cot(90° − z) arcminutes near the horizon.
Zenith Angle Dependence
Refraction is zero for a ray arriving directly overhead (z = 0°) because it travels straight down through horizontal layers without encountering an angled boundary. As the zenith angle increases toward 90° (horizon), the ray travels a much longer slanted path through the atmosphere, crossing many more layers at a shallow angle, so cumulative bending increases dramatically. At z = 45° the displacement is only about 1 arcminute, but at the horizon (z ≈ 90°) it reaches approximately 34–35 arcminutes — just over half a degree, which equals the apparent diameter of the Sun or Moon.
Temperature and Pressure Corrections
The standard refraction formula assumes average atmospheric conditions (pressure P₀ = 101.325 kPa, temperature T₀ = 10 °C). Real conditions vary: cold, dense air refracts more strongly; warm, less dense air refracts less. Corrected refraction is scaled by the factor (P/P₀) × (T₀/T) where pressures and temperatures are in SI units. On a hot summer day the atmosphere refracts light less than on a cold winter morning, which is why mirages are more common in summer over hot surfaces — temperature inversions create layers where refraction bends light dramatically.
Mirages as Extreme Refraction
A mirage occurs when a strong temperature gradient (and thus refractive index gradient) bends light so severely that rays curve upward from the ground to your eye, creating an inverted image of the sky that looks like a pool of water. In an inferior mirage (hot road), a layer of hot air near the surface has a lower refractive index than cooler air above, causing the light ray to bend upward. In a superior mirage (cold arctic sea), a cold layer near the surface has higher refractive index, bending rays downward and making distant objects appear elevated or floating above the horizon.
The Key Equation
- R — Atmospheric refraction (arcminutes): The angular amount by which the apparent position of a celestial object is shifted upward relative to its true geometric position.
- a — Apparent altitude (degrees (°)): The observed angle of the object above the horizon as seen by the observer, before correcting for refraction.
- \tan — Tangent function (dimensionless): Trigonometric function relating the angle to the ratio of opposite to adjacent sides; its small value near the horizon causes R to become large there.
Worked Example
Problem. A star is observed at an apparent altitude of a = 5° above the horizon. Calculate the atmospheric refraction correction R using Bennett's formula, and hence determine the true altitude of the star.
Answer. The atmospheric refraction is approximately 9.7 arcminutes, so the star's true altitude is about 4.84°, meaning it is actually almost 10 arcminutes lower in the sky than it appears.
Origins · 1 minIbn al-Haytham (Alhazen) and how this was discovered
Ibn al-Haytham was the first scholar to correctly explain atmospheric refraction in his monumental work Kitāb al-Manāẓir (Book of Optics, c. 1021). He argued that the atmosphere has measurable thickness and that its varying density bends light rays, causing stars and the Sun to appear displaced from their true positions. He estimated the height of the atmosphere by observing the duration of twilight and using geometry, arriving at a value that is remarkably consistent with modern estimates for the optically significant lower atmosphere. Ibn al-Haytham is also credited with inventing the camera obscura (pinhole camera) and was perhaps the first to insist that scientific claims must be tested by repeatable experiment — a foundational principle of modern science — centuries before the European Scientific Revolution.
Related Concepts
- Snell's Law of Refraction — Atmospheric refraction is governed by Snell's Law applied continuously across layers of varying refractive index; understanding n₁ sin θ₁ = n₂ sin θ₂ is essential to deriving refraction corrections.
- Ray Model of Light — Atmospheric refraction is analysed using the ray model, tracing straight-line segments through each atmospheric layer and applying bending rules at each boundary.
- Total Internal Reflection — Total internal reflection is the extreme limit of refraction when a ray cannot pass into a less dense medium; understanding both phenomena deepens knowledge of how light behaves at refractive index boundaries.
- Dispersion and Prisms — Because refractive index varies with wavelength, atmospheric refraction is slightly different for each colour of light, producing phenomena such as the green flash and chromatic aberration in astronomical observations.