Seeing beyond the horizon

That’s Maths: Light travels more easily through air of lower density

From a hilltop, the horizon lies below the horizontal level at an angle called the “dip”. Around 1020 AD, the brilliant Persian scholar al-Biruni used a measurement of the dip, from a mountain of known height, to get an accurate estimate of the size of the Earth. It is claimed that his estimate was within 1 per cent of the true value but, since he was not aware of atmospheric refraction and made no allowance for it, this high precision must have been fortuitous.

From a standpoint by the ocean shore, the horizon appears to be about 5km away. From an elevation of 100m, it is some 36km and from a mountain peak at 500m, about 80km. There is a simple formula for the distance: d = (2RH/1,000). R is the Earth’s radius (6,370 km) and H is the elevation in metres.

This formula gives the distance in kilometres to the geometric horizon, but the visible horizon may be closer or farther away: the transparency of the atmosphere, which allows light to pass through it without scattering, depends upon prevailing meteorological conditions, and variations in density cause light to deviate from a straight path.

Visibility

In extremely clean, clear air, the visibility can be over 200km, but this is rare. Visibility is normally limited by turbulence, humidity and pollution and is usually no greater than about 80km. The ideal conditions are a crisp, cold day with still air, low humidity and an absence of pollution. The colder the air, the less humidity it can hold.

January 2021 was unusually cold, with air temperatures across the country well below average. On January 12th, photographer Niall O'Carroll took a remarkable photograph from Howth Head, showing the mountains of north Wales, with Snowdon prominent in centre-shot.

Using the above formula, the horizon from the top of Snowdon, at 1,085m, is 118km. The horizon from the Ben of Howth (171m) is at 47km. The sum of these distances (165km) exceeds the great circle distance of 140km between the two peaks. Therefore, each is above the geometric horizon of the other. However, the photograph shows several lesser peaks in Wales, and extensive lower ground. How can this be?

Refraction

Atmospheric refraction is the bending of light from a straight line as it passes through the atmosphere. There are several factors influencing refraction temperature, pressure and humidity amongst them but the most important is the decrease of density with height. Normally, the air temperature drops with increasing altitude mountain tops are colder than lower ground. But in an inversion, cold air underlies warmer air.

Air is compressible and its density decreases exponentially with height. Light travels more easily through air of lower density. The route taken between two points by a light ray is the path that can be traversed in the least time.

It is easier for light to travel along a route that passes upward through thinner air. As a result, the transition from cold dense air to thinner warmer air bends the light, and objects below the (geometric) horizon can become visible: in suitable weather conditions, we can actually see “beyond the horizon”.

Peter Lynch is emeritus professor at UCD School of Mathematics and Statistics. He blogs atthatsmaths.com

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