Editing Horizon (section)

== Effect of atmospheric refraction ==
{{Further|Atmospheric refraction}}
{{See also|Line-of-sight propagation#Atmospheric refraction{{!}}Effective Earth radius}}
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Due to [[atmospheric refraction]] the distance to the visible horizon is further than the distance based on a simple geometric calculation. If the ground (or water) surface is colder than the air above it, a cold, dense layer of air forms close to the surface, causing light to be refracted downward as it travels, and therefore, to some extent, to go around the curvature of the Earth. The reverse happens if the ground is hotter than the air above it, as often happens in deserts, producing [[mirage]]s. As an approximate compensation for refraction, surveyors measuring distances longer than 100 meters subtract 14% from the calculated curvature error and ensure lines of sight are at least 1.5 metres from the ground, to reduce random errors created by refraction.
[[File:Pale Horizon.jpg|thumb|right|Typical desert horizon]]

If the Earth were an airless world like the Moon, the above calculations would be accurate. However, Earth has an [[Earth's atmosphere|atmosphere of air]], whose [[density]] and [[refractive index]] vary considerably depending on the temperature and pressure. This makes the air refract light to varying extents, affecting the appearance of the horizon. Usually, the density of the air just above the surface of the Earth is greater than its density at greater altitudes. This makes its refractive index greater near the surface than at higher altitudes, which causes light that is travelling roughly horizontally to be refracted downward.<ref name="ProctorRanyard1892">{{cite book|last1=Proctor|first1=Richard Anthony|last2=Ranyard|first2=Arthur Cowper|title=Old and New Astronomy|url=https://archive.org/details/oldnewastronomy00procuoft|year=1892|publisher=Longmans, Green and Company|pages=[https://archive.org/details/oldnewastronomy00procuoft/page/73 73]}}</ref> This makes the actual distance to the horizon greater than the distance calculated with geometrical formulas. With standard atmospheric conditions, the difference is about 8%. This changes the factor of 3.57, in the metric formulas used above, to about 3.86.<ref name="ATYoungDistToHoriz"/> For instance, if an observer is standing on seashore, with eyes 1.70 m above sea level, according to the simple geometrical formulas given above the horizon should be 4.7&nbsp;km away. Actually, atmospheric refraction allows the observer to see 300 metres farther, moving the true horizon 5&nbsp;km away from the observer.

This correction can be, and often is, applied as a fairly good approximation when atmospheric conditions are close to [[Reference atmospheric model|standard]]. When conditions are unusual, this approximation fails. Refraction is strongly affected by temperature gradients, which can vary considerably from day to day, especially over water. In extreme cases, usually in springtime, when warm air overlies cold water, refraction can allow light to follow the Earth's surface for hundreds of kilometres. Opposite conditions occur, for example, in deserts, where the surface is very hot, so hot, low-density air is below cooler air. This causes light to be refracted upward, causing [[mirage]] effects that make the concept of the horizon somewhat meaningless. Calculated values for the effects of refraction under unusual conditions are therefore only approximate.<ref name="ATYoungDistToHoriz" /> Nevertheless, attempts have been made to calculate them more accurately than the simple approximation described above.

Outside the visual wavelength range, refraction will be different. For [[radar]] (e.g. for wavelengths 300 to 3&nbsp;mm i.e. frequencies between 1 and 100&nbsp;GHz) the radius of the Earth may be multiplied by 4/3 to obtain an effective radius giving a factor of 4.12 in the metric formula i.e. the radar horizon will be 15% beyond the geometrical horizon or 7% beyond the visual. The 4/3 factor is not exact, as in the visual case the refraction depends on atmospheric conditions.

;Integration method—Sweer
If the density profile of the atmosphere is known, the distance ''d'' to the horizon is given by<ref name="Sweer1938">{{cite journal|author=Sweer, John |bibcode=1938JOSA...28..327S |title=The Path of a Ray of Light Tangent to the Surface of the Earth|journal=Journal of the Optical Society of America|volume=28 |date=1938|issue=9 |pages=327–329| doi=10.1364/JOSA.28.000327
 }}</ref>

:<math>d={{R}_{\text{E}}}\left( \psi +\delta  \right) \,,</math>

where ''R''<sub>E</sub> is the radius of the Earth, ''ψ'' is the dip of the horizon and ''δ'' is the refraction of the horizon. The dip is determined fairly simply from

:<math>\cos \psi = \frac{{R}_{\text{E}}{\mu}_{0}}{\left( {{R}_{\text{E}}}+h \right)\mu } \,,</math>

where ''h'' is the observer's height above the Earth, ''μ'' is the index of refraction of air at the observer's height, and ''μ''<sub>0</sub> is the index of refraction of air at Earth's surface.

The refraction must be found by integration of

:<math>\delta =-\int_{0}^{h}{\tan \phi \frac{\text{d}\mu }{\mu }} \,,</math>

where <math>\phi\,\!</math> is the angle between the ray and a line through the center of the Earth. The angles ''ψ'' and <math>\phi\,\!</math> are related by

:<math>\phi =90{}^\circ -\psi \,.</math>

;Simple method—Young
A much simpler approach, which produces essentially the same results as the first-order approximation described above, uses the geometrical model but uses a radius {{nowrap|1=''R′'' = 7/6 ''R''<sub>E</sub>}}. The distance to the horizon is then<ref name="ATYoungDistToHoriz"/>

:<math>d=\sqrt{2 R^\prime h} \,.</math>

Taking the radius of the Earth as 6371&nbsp;km, with ''d'' in km and ''h'' in m,

:<math>d \approx 3.86 \sqrt{h} \,;</math>

with ''d'' in mi and ''h'' in ft,

:<math>d \approx 1.32 \sqrt{h} \,.</math>

In the case of [[radar]] one typically has {{nowrap|1=''R′'' = 4/3 ''R''<sub>E</sub>}} resulting (with ''d'' in km and ''h'' in m) in

:<math>d \approx 4.12 \sqrt{h} \,;</math>

Results from Young's method are quite close to those from Sweer's method, and are sufficiently accurate for many purposes.